Introductory Physics Lab1

Circular Motion

In this lab, you are going to play with a toy. Here is a diagram of this toy.

Basically, this is a mass on a string attached to a rubber stopper. The string passes through a glass tube.There is a piece of tape to help you keep the length of the string outside the glass tube constant. The way youuse this toy is to spin the rubber stopper around in a circle. Here is an example:

alhoson coll

So, what are you going to do? First, the physics. If I were to draw free body diagram for the stopper at theabove instant, it would look like this:

There are only two forces on the stopper, the tension from the string and the gravitational force. The netforce is NOT zero. This is because the stopper is accelerating. It is moving at a constant speed, but changingdirection. When an object moves in a circle, it has a magnitude of acceleration:

The direction of the acceleration is towards the center of the circle. If you want to know where this formulacomes from, check this out. So, for the above instant, the acceleration would be in the negative x-direction. IfI write down the force equation for the x- and y-direcitons, I get (note that hte acceleration in the y-directionis zero m/s2).

(note - this mass is the mass of the stopper) What can you measure and what can you solve for? First, thereis a relationship between r and L:

You can measure: - The tension ( T ). This is just the mass times g (the mass of the hanging mass).- The length ( L ). Remember, this is the distance from the point of rotation to the center of the stopper.

- The period of the rotation - how long it takes to complete one revolution. This will be easiest to measure bytiming say 10 rotations and dividing by 10.

From the period, you can find the velocity:

What about theta? Is there anyway to get the angle from the diagram? Yes. Just look at the y-direction andsolve for theta:

I am not going to put all this together for you. But, let me say this. If you keep the mass on the bottom of thestring (M2) constant, then there is a relationship between the length of the string and the period - I am notsaying it is a linear relationship. So, here is what you are going to do:

- Pick a mass to hang on the end of the string. If you pick something really high, you are going to have toswing that stopper really fast and bad things may happen.- For that mass, pick a length of string to be hanging out of the string holder. You can keep track of what thislength is by putting a piece of tape at the bottom of the string holder (glass rod). Make sure while you areswinging the stopper that the tape stays stationary but does not touch the glass.- Get the thing swinging.- The swinging person should only concentrate on making sure the tape stays constant- Someone needs to count 10 or so periods and time it. Repeat this process 5 times to get an uncertainty.- Change the length of the string and do it again.- Do this for 5 different lengths of string.- Make an a graph that shows a linear function (it will involve the period and the length). Find the slope ofthis linear graph and figure out what it means.

Extra stuff: - Calculate what your angle is. If you want to check it, you can take a picture of the stopper swinging (I canhelp you with this).

Free fall

Objectives

Acceleration is the rate at which the velocity of an object changes over time.An object’s acceleration is the result of the sum of all the forces acting onthe object, as described by Newton’s second law. Under ideal circumstances,gravity is the only force acting on a freely falling object. In this lab, youwill measure the displacement of a freely falling object, calculate the averagevelocity of a falling object at set time intervals, and calculate the object’sacceleration due to gravity. The objectives of this experiment are as follows:

1. to measure the displacement of a freely falling object,

2. to test the hypothesis that the acceleration of a freely falling object isuniform,

3. to calculate the uniform acceleration of a falling object due to gravity,g.

Theory

The instant when the ball is released is considered to be the initial timet = 0. The position of the ball along the ruler is described by the variabley. The position of the ball at a time t is given by

y(t) = y0 + v0t +1

2gt2. (1)

If the ball is released from rest, the initial velocity is zero: v0 = 0.Therefore,

y(t) = y0 +1

2gt2. (2)

1

Accepted values

The acceleration due to gravity varies slightly, depending on the latitudeand the height above the earths surface. In this experiment the change inheight of the falling object is negligible and can be approximated as 0 km forits entire descent. The acceleration due to gravity at 40◦52′21′′ N latitude(the latitude of Lehman College) and 0 km altitude is

g = 9.802 m/s2. (3)

Apparatus

The setup, depicted in Fig. 1, is composed of the following parts:

• electromagnet,

• steel ball,

• ruler,

• mobile photogate,

• timer,

• power supply,

• paper cup.

The power supply provides an output of 5 V to an electromagnet. Whenthe switch is in the on position, the electromagnet can hold the steel ballunder it. Once the timer is set to the off position, current stops circulatingthrough the electromagnet, and the ball starts falling.

The sudden change in the current circulating through the magnet pro-duces, following Lenz’s law, a short current peak that propagates throughthe red wire in Fig. 1. Part of this wire is placed in parallel to the wireattaching the unused photogate to the timer (blue wire in Fig. 1). Thecurrent in the blue wire produces a magnetic field around it. The red wire,when sufficiently close to the blue one, is affected by this magnetic field,which induces a current on it. This current, in the form of a short peak, isinterpreted by the timer as an interruption of the photogate, triggering thetimer.

Using these principles, the setup allows to have a precise account ofthe initial time, since the timer starts counting when the ball is released.The second trigger of the timer happens when the ball goes through thephotogate. In this moment, the timer stops counting. Therefore, the timerindicates the time (in seconds) it took the ball to go from the top positionto the photogate.

2

Moving the photogate to different heights and measuring the time theball takes to fall will provide the information necessary to measure the ac-celeration of gravity.

magnet clamp

photogate clamp

electromagnet

ball

vertical rod

paper cup

ruler

0:2791

power supply

switchtimer

photogate

Figure 1: Experimental setup.

Procedure

1. Adjust the top clamp (the one holding the magnet) in such a way that,with the ruler standing on the table, the center of the ball is at aboutthe same height as the zero of the ruler (see Fig. 2);

2. turn on the timer by moving the switch to the pulse mode;

3. adjust the height of the bottom clamp (the one holding the photogate)to around 10 cm below the magnet;

4. align the photogate with the electromagnet so that the ball will passthrough the photogate while falling. To do so, you can rotate theclamp around the vertical rod, and adjust the photogate along thehorizontal rod. To check that the alignment is correct, hold the topof the ruler right below the magnet so that it doesn’t touch the table,and make sure that the ruler goes through the photogate (see Fig. 3);

3

Figure 2: Setup for the mag-net holder.

Figure 3: Alignment of thephotogate.

5. measure the position of the photogate, and record it on the table as avalue for y;

6. switch on the magnet, and place the ball under it, making sure thatit remains there;

7. hold a paper cup right below the photogate to catch the ball when itfalls;

8. while paying attention to the timer, switch the magnet off. The ballwill fall. Three outcomes are possible:

(a) the timer starts and stops immediately, showing a really smallvalue (like 0.0001). In this case, disregard this value and measureagain,

(b) the timer doesn’t start when the magnet is switched off, but itstarts later when the ball goes through the photogate. Therefore,the timer keeps running after the ball has fallen. In this case,press reset and measure again,

(c) the timer starts when the magnet is switched off, and stops whenthe ball goes through the photogate. In this case, record the timeon the table as a value for t;

4

9. move the photogate to a position around 10 cm below the previousposition;

10. measure again: repeat steps 4 to 9 until there is no more space to keepthe paper cup under the photogate (around 80 cm).

Troubleshooting

If the timer never starts when the switch is changed to the off position,it could be due to several reasons. First, check that when it is in the onposition, the electromagnet is able to hold the ball. If this is not the case,it is possible that there is a short in the circuit. Turn the power supply offand ask your instructor for help. If the magnet is able to hold the ball, butthe timer doesn’t start when switched off, a possible solution is to connectthe red and black power cables to the front of the power supply, rather thanto the back, and selecting a bit higher voltage (around 6 V).

Data

5

6

Abstract

In this experiment, we tested Newton’s second law and determined the acceleration due to gravity on the Moon. We used a frictionless air table equipped with a spark timer, and studied the motion of a puck attached to a vertically hanging weight. In both setups that we tested, the puck’s acceleration changed linearly with the mass of hanging weight as expected from Newton’s second law, but in one case, the measured 𝑔𝑔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 value was not consistent with the known value within the uncertainty.

force exp

Introduction

An object which is acted upon a non-zero total force undergoes an acceleration in the direction of the force. The relation between the acceleration of the object and the net force acting on it is given by Newton’s second law [1], which can be expressed as

�⃗�𝐹𝑚𝑚𝑛𝑛𝑛𝑛 = 𝑚𝑚�⃗�𝑎 (1)

where �⃗�𝐹𝑚𝑚𝑛𝑛𝑛𝑛 is the net force, m is the object’s mass and �⃗�𝑎 denotes the object’s acceleration. As an example, for the case of free fall on Earth, �⃗�𝐹𝑚𝑚𝑛𝑛𝑛𝑛 is equal to the object's weight and Newton’s second law reads

𝐹𝐹𝑚𝑚𝑛𝑛𝑛𝑛 = 𝑚𝑚𝑔𝑔 = 𝑚𝑚𝑎𝑎. (2) From this we can conclude that the object falls on Earth at a constant rate 𝑎𝑎 = 𝑔𝑔 = 9.81 𝑚𝑚/𝑠𝑠2.

The goal of this experiment was to test Newton’s second law on the Moon by studying the motion of a puck attached to hanging weights on a frictionless air table, and to measure the acceleration due to Moon’s gravity 𝑔𝑔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚. In the first investigation, the motion of the puck pulled by three different weights was analyzed. The position of the puck was measured periodically, and the velocity and acceleration were determined. The accelerations were compared to the mass of the system to test Newton’s second law and to determine 𝑔𝑔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚. The second investigation involved the same procedure, but the mass of the puck was doubled by adding extra weights to it. A similar analysis was performed and a second measured value of 𝑔𝑔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 was obtained.

Investigation 1

The setup consists of pucks that float on a flat glass surface table using compressed air [2]. The motion of the pucks can be tracked from dots that are periodically marked by a spark timer on a white paper placed under the pucks. A sheet of carbon paper placed under the white paper provides the electrical connection between the pucks and the spark timer. A pulley hooked up to the side of the table allows a puck to be attached to a vertically moving mass hanger.

Before data were collected, the air table was leveled by adjusting its three adjustable legs. The spark timer was set to 10 𝐻𝐻𝐻𝐻 so that, when the circuit was closed by pressing the associated switch pedal, sparks were generated every 0.1 𝑠𝑠. The mass of the puck was measured to be 𝑚𝑚𝑝𝑝 = 550 𝑔𝑔. For the first trial, a mass of 𝑚𝑚𝑤𝑤 = 200 𝑔𝑔 was used as the hanging weight.

Data were collected by pressing down the spark timer pedal and then releasing the puck. The puck slid along the air table, generating sparks at fixed time intervals. When the puck reached the end of the table, the pedal was released and the white paper was removed and inspected. After skipping the first few dots, ten consecutive dots for which the puck was in a smooth motion were labeled 1-10. The displacement between each set of three adjacent dots, 𝛥𝛥𝛥𝛥, was measured with a ruler and recorded. Displacement #1, for example, was the distance between dots 1 and 3, while displacement #2 was the distance between dots 2 and 4 (and so on). The instrumental uncertainty in these measurements, 𝛿𝛿𝛥𝛥𝛥𝛥 =0.3 𝑐𝑐𝑚𝑚, was assessed based on the radius of the spark timer dots. The time interval between each pair of adjacent dots, 𝛥𝛥𝛥𝛥 = 0.1 𝑠𝑠, was also noted. The relative uncertainty 𝛿𝛿𝛥𝛥𝛥𝛥/𝛥𝛥𝛥𝛥 = 0.2% was provided by lab personnel. After these values were gathered and recorded for the 200 𝑔𝑔 trial, the process was repeated two more times with hanging masses of 400 and 800 𝑔𝑔, respectively. The resulting data are provided in Table 1 below.

Table 1: Displacement, time, and velocity data for the puck with different hanging masses of 200 𝑔𝑔, 400 𝑔𝑔 and 800 𝑔𝑔 respectively. The mass of the puck is 𝑚𝑚𝑝𝑝 = 550 𝑔𝑔.

mw 200 g Interval Δx (cm) t (s) v (cm/s) δv (cm/s)

1-3 2.6 0.1 13.0 1.5 2-4 3.3 0.2 16.5 1.5 3-5 4.4 0.3 22.0 1.5 4-6 5.2 0.4 26.0 1.5 5-7 6.2 0.5 31.0 1.5 6-8 7.1 0.6 35.5 1.5 7-9 7.9 0.7 39.5 1.5

8-10 8.7 0.8 43.5 1.5

mw 400 g Interval Δx (cm) t (s) v (cm/s) δv (cm/s)

1-3 4.0 0.1 20.0 1.5 2-4 5.6 0.2 28.0 1.5 3-5 6.9 0.3 34.5 1.5 4-6 8.3 0.4 41.5 1.5 5-7 9.5 0.5 47.5 1.5 6-8 11.1 0.6 55.5 1.5 7-9 12.4 0.7 62.0 1.5

8-10 13.6 0.8 68.0 1.5

mw 800 g Interval Δx (cm) t (s) v (cm/s) δv (cm/s)

1-3 5.7 0.1 28.5 1.5 2-4 7.8 0.2 39.0 1.5 3-5 9.7 0.3 48.5 1.5 4-6 11.4 0.4 57.0 1.5 5-7 13.5 0.5 67.5 1.5 6-8 15.6 0.6 78.0 1.5 7-9 17.2 0.7 86.0 1.5

8-10 19.2 0.8 96.0 1.5

In order to determine the acceleration of the puck, the velocity had to be calculated from the raw displacement and time data at various points. Because each displacement 𝛥𝛥𝛥𝛥 was defined over two time intervals 𝛥𝛥𝛥𝛥, the average velocity 𝑣𝑣 during each displacement was calculated as:

𝑣𝑣 = ∆𝛥𝛥2∆𝛥𝛥

(3)

The propagated uncertainties in these velocities, 𝛿𝛿𝑣𝑣, are given by the relative uncertainties in time and displacement as:

𝛿𝛿𝑣𝑣𝑣𝑣

= ��𝛿𝛿∆𝛥𝛥∆𝛥𝛥

�2

+ �𝛿𝛿∆𝛥𝛥∆𝛥𝛥

�2

(4)

The average velocities for each displacement interval were assumed to take place at the middle of each associated time interval. In other words, for the first 𝑣𝑣 value, corresponding time was 0.1 𝑠𝑠, for the second one, 𝛥𝛥 was 0.2 𝑠𝑠 and so on. The derived 𝑣𝑣, 𝛿𝛿𝑣𝑣 and 𝛥𝛥 values are also included in Table 1.

To calculate the acceleration, a plot of the average velocities vs. time plot was created (see Fig. 1 below). The data points were input into the IPL straight-line fit calculator [3] to determine the slopes, 𝑎𝑎, and their uncertainties, 𝛿𝛿𝑎𝑎. The resulting values are provided in Table 2 below.

Fig. 1: The puck’s velocity as a function of time. The data for 𝑚𝑚𝑤𝑤 = 200 𝑔𝑔, 400 𝑔𝑔 and 800 𝑔𝑔 are shown in green, red and orange respectively.

As previously discussed, Newton’s second law states that the sum of all forces on an object is equal to the product of its mass and acceleration. If we denote the mass of the hanging weight and puck as, respectively, 𝑚𝑚𝑤𝑤 and 𝑚𝑚𝑝𝑝 , then Eq. (1) can be used to describe the acceleration of the entire puck & hanging mass assembly, 𝐹𝐹 = �𝑚𝑚𝑤𝑤 + 𝑚𝑚𝑝𝑝�𝑎𝑎. The force on this assembly is just the gravitational force on the hanging mass, given by 𝐹𝐹 = 𝑚𝑚𝑤𝑤𝑔𝑔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚. The combination of these two equations gives:

𝑎𝑎 =𝑚𝑚𝑤𝑤

𝑚𝑚𝑤𝑤 + 𝑚𝑚𝑝𝑝𝑔𝑔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑟𝑟 𝑔𝑔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 (5)

where 𝑟𝑟 = 𝑚𝑚𝑤𝑤/(𝑚𝑚𝑤𝑤 + 𝑚𝑚𝑝𝑝) is the ratio of the attached mass to the total mass of the puck + attached weight system.

Thus there should be a linear relationship between the acceleration and the mass ratio 𝑟𝑟. In order to determine the acceleration due to gravity, the mass ratio was calculated for each trial (Table 2) and plotted as the independent variable against the acceleration (Fig. 2). Since the masses were measured very precisely, the errors in the mass ratios r were neglected. As expected from Newton’s second law, the data agrees well with a linear trend line and based on Eq. (6).

y = 44.524x + 8.339R² = 0.998

y = 68.452x + 13.821R² = 0.999

y = 96.012x + 19.357R² = 0.999

0.0

20.0

40.0

60.0

80.0

100.0

120.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

v (c

m/s

)

t (s)

Puck's velocity vs. time

Table 2: Acceleration of the puck and mass ratio data for 𝑚𝑚𝑝𝑝 = 550 𝑔𝑔.

mw (g) r a (cm/s^2) δa (cm/s^2) 200 0.267 44.5 2.3 400 0.421 68.5 2.3 800 0.593 96.0 2.3

Fig. 2: Puck’s acceleration as a function of mass ratio 𝑟𝑟.

According to Eq. (5), the slope of this trend line is expected to be equal to 𝑔𝑔𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚. By inputting the data into the IPL calculator, the slope of the best fit line was determined to be 158 𝑐𝑐𝑚𝑚/𝑠𝑠2. The uncertainty in the slope was 10 𝑐𝑐𝑚𝑚/𝑠𝑠2.Thus, the acceleration due to the Moon’s gravity was experimentally determined to be 1.58 ± 0.10 𝑚𝑚/𝑠𝑠2. The expected value, 1.62 𝑚𝑚/𝑠𝑠2, falls within the range around the measured value defined by the measured value’s uncertainty, so Newton’s second law was verified.

Investigation 2

The second investigation consisted of the same setup as Investigation 1, except in this instance extra weight was added on the puck to double its mass to 𝑚𝑚𝑝𝑝 = 1100 𝑔𝑔. The same procedure was used as before; the puck was attached to three different hanging masses and released. From the dots on the white paper, the displacements were measured and recorded in Table 3 below.

y = 158.054x + 2.214R² = 1.000

30.0

40.0

50.0

60.0

70.0

80.0

90.0

100.0

110.0

0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 0.650

a (c

m/s

^2)

r

Puck's acceleration vs. mass ratio

Table 3: Displacement, time, and velocity data for the puck with different hanging masses of 200 𝑔𝑔, 400 𝑔𝑔 and 800 𝑔𝑔 respectively. The mass of the puck is 𝑚𝑚𝑝𝑝 = 1100 𝑔𝑔.

mw 200 g Interval Δx (cm) t (s) v (cm/s) δv (cm/s)

1-3 1.5 0.1 7.5 1.5 2-4 2.1 0.2 10.5 1.5 3-5 2.6 0.3 13.0 1.5 4-6 3.1 0.4 15.5 1.5 5-7 3.6 0.5 18.0 1.5 6-8 4.2 0.6 21.0 1.5 7-9 4.6 0.7 23.0 1.5

8-10 5.1 0.8 25.5 1.5

mw 400 g Interval Δx (cm) t (s) v (cm/s) δv (cm/s)

1-3 2.6 0.1 13.0 1.5 2-4 3.6 0.2 18.0 1.5 3-5 4.2 0.3 21.0 1.5 4-6 5.2 0.4 26.0 1.5 5-7 6.0 0.5 30.0 1.5 6-8 7.0 0.6 35.0 1.5 7-9 7.7 0.7 38.5 1.5

8-10 8.7 0.8 43.5 1.5

mw 800 g Interval Δx (cm) t (s) v (cm/s) δv (cm/s)

1-3 4.1 0.1 20.5 1.5 2-4 5.2 0.2 26.0 1.5 3-5 6.6 0.3 33.0 1.5 4-6 7.8 0.4 39.0 1.5 5-7 9.3 0.5 46.5 1.5 6-8 10.4 0.6 52.0 1.5 7-9 11.8 0.7 59.0 1.5

8-10 13.0 0.8 65.0 1.5

Once again, the average velocities and their uncertainties were calculated using Eqs. (3) and (4), respectively. The times at which these velocities took place were determined from the elapsed time at the central dot of each consecutive triplet. Velocity vs. time was graphed for each of the three weights used, as shown in Fig. 3.

The accelerations for each case (and their uncertainties) were determined from the IPL straight-line fit calculator and are displayed in Table 4, along with their associated mass ratio. The accelerations of the pucks were again graphed against the mass ratio (Fig. 4) and error bars and a linear trend line were added.

Fig. 3: Puck’s velocity as a function of time. The data for 𝑚𝑚𝑤𝑤 = 200 𝑔𝑔, 400 𝑔𝑔 and 800 𝑔𝑔 are shown in green, red and orange respectively.

Table 4: Acceleration of the puck and mass ratio data for 𝑚𝑚𝑝𝑝 = 1100 𝑔𝑔.

mw (g) r a (cm/s^2) δa (cm/s^2) 200 0.154 25.6 2.3 400 0.267 43.1 2.3 800 0.421 64.4 2.3

Based on Eq. (6), the data were expected to once again linearly change with respect to mass ratio 𝑟𝑟, and the slope of this plot was expected to be equal to the acceleration due to gravity. Although the linear trend line agreed very well with the data, the resulting slope value, 1.44 ± 0.12 𝑚𝑚/𝑠𝑠2, was not consistent with the expected value, 1.62 𝑚𝑚/𝑠𝑠2. As such, Newton’s second law was not corroborated by this investigation.

There might be many reasons for this slight disagreement. It is possible that with the added weight, the puck was not able to float perfectly on the table but perhaps touched the paper underneath which created friction. The pulley could also affect the results by adding further friction. If present, this systematic error would have resulted in a force opposite the puck’s direction of motion, which would decrease the measured acceleration, which is what we observed.

y = 25.595x + 5.232R² = 0.999

y = 43.095x + 8.732R² = 0.998

y = 64.405x + 13.643R² = 0.999

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

v (c

m/s

)

t (s)

Puck's velocity vs. time

Fig. 4: Puck’s acceleration as a function of mass ratio r.

Conclusion

This experiment consisted of two investigations, both of which attempted to test Newton’s second law and determine the gravitational acceleration on the Moon. An air table was used to create a frictionless surface and the motion of a puck attached to hanging masses was periodically marked by a spark timer. From these marks, the velocity of the puck at consecutive points in time was calculated. By plotting these velocities against the moments of times at which they occurred, the accelerations were determined. Using Newton’s second law, these accelerations were shown to be linearly related to the ratio of the attached weight to the total weight of the puck + weight system. By plotting these two quantities against each other, the linear relationship was tested and the acceleration due to Moon’s gravity was determined.

In both investigations, the acceleration data showed a linear trend with respect to the mass ratio as expected. In the first investigation, the gravitational acceleration was measured to be 1.58 ± 0.10 𝑚𝑚/𝑠𝑠2, which was consistent with the expected value of 1.62 𝑚𝑚/𝑠𝑠2 considering the stated uncertainty. However in the second investigation, the known value did not fall within the measured acceleration value of 1.44 ± 0.12 𝑚𝑚/𝑠𝑠2. It is possible that the air table did not create enough of an air cushion to completely eliminate the effect of friction, which would lower the measured value. It is possible that the pulley used in the setup also added some friction.

In order to mitigate the effect of the unaccounted-for errors discussed above, further efforts could be made to reduce the friction present in the system. The air pressure could be increased to create a better air cushion and the pulley wheel could be better lubricated. To improve the precision of the results, several changes to the procedure could be made. While the uncertainties in the time intervals were very small (and difficult to reduce further), the position measurements were significantly limited by the size of the spark timer dots. Thus, the precision with which the acceleration due to gravity is determined could be

y = 144.776x + 3.754R² = 0.999

20

25

30

35

40

45

50

55

60

65

70

0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450

a (c

m/s

^2)

r

Puck's acceleration vs. mass ratio

improved by decreasing 𝛿𝛿𝛥𝛥𝛥𝛥. This could potentially be achieved by upgrading the spark timer or determining the center of the dots in a more precise manner.

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Title:  Motion  of  a  Projectile  

Abstract:  In  this  experiment  the  motion  of  a  projectile  was  evaluated.  A  marble  was  launched  horizontally  off  a  table-­‐top  and  the  distance  it  traveled  forward  was  measured.  The  equations  of  motion  for  a  horizontal  projectile  were  used  to  calculate  the  velocity  of  the  projectile  as  it  left  the  table-­‐top.    Using  this  information,  the  height  of  the  table  was  changed  and  the  marbles  were  launched  a  second  time.  Using  the  previously  determined  initial  velocity,  which  should  be  unchanged,  e  new  travel  distance  was  calculated  and  compared  with  the  measured  distance.    The  experimental  results  confirmed  this  to  be  the  case  as  had  been  hypothesized.            

Introduction:  

Background:  The  question  of  projectile  motion  is  one  that  has  been  worked  on  for  many  centuries.  Aristotle  believed  all  motion  was  linear,  and  that  in  order  to  continue  to  move  an  object  must  continue  to  have  force,  or  impetus  behind  it.  In  the  17th  18th  centuries  Galileo,  Newton  and  others  applied  the  “scientific  method”  and  gathered  experimental  data  to  debunk  Aristotle’s  ideas,  thereby  developing  the  modern  classical  theories  of  motion.  These  laws  of  motion  are  based  on  a  set  of  kinematic  equations  the  x  (horizontal)  and  y  (vertical)  directions.  

(1) x  =  x0  +v0t+ ½  at2  ,  (2)  v  =  v0+  at,  (3)  v2  =  v02  +2a(x  -­‐  x0)

For  horizontally  launched  projectiles,  these  equations  are:  

(1) y  =  ½  at2  ,  (2)  vy  =  v0y  +  gt,  (3)  vy  =  (2gy)½  (4)  x  =  v0xt

These  equations  will  be  used  to  calculate  the  initial  velocity  of  the  projectile,  v0x  ,  and  to  calculate  the  distance,  x,  traveled  when  the  table  height  was  changed.  

Objective:  By  launching  a  projectile  horizontally  the  validity  of  the  equation  of  motions  will  be  tested.        

Hypothesis:  The  equations  of  motion  predict  that  when  a  projectile  is  launched  at  greater  heights  the  distance  it  travels  forward  will  increase.  

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Materials  and  Methods:      

Materials  for  this  lab  included  a  marble,  a  stopwatch,  a  ramp,  a  table,  a  chair,  corn  starch,  water  a  tape  measure  a  plumb  line  and  construction  paper.  

The  procedure  was  to  set  up  a  ramp  on  a  table  and  use  the  plumb  bob  to  make  sure  that  the  table  surface  is  horizontal.    The  marbles  are  released  from  a  fixed  point  at  the  top  of  the  ramp  and  allowed  to  accelerate  to  a  velocity  upon  leaving  the  table.    The  impression  of  the  corn-­‐starched  coated  marble  was  noted  on  the  construction  paper  placed  on  the  floor  to  obtain  the  horizontal  distance  traveled.    

Several  trials  are  made  and  the  heights  and  distanced  traveled  forward  are  documented  in  the  Data  Table.    This  is  repeated  at  a  second  table  height.  

The  first  data  set  is  used  to  calculate  the  initial  velocity  of  the  projectile  leaving  the  table.    This  velocity  is  used  with  the  second  data  set  to  predict  the  horizontal  distance  traveled  by  the  marble  and  compare  with  experimental  results.    

Data  and  Results:  

-­‐-­‐-­‐  Data  Table  here.    Title  above  each  table;  e.g.  Procedure  1:  Table  ht  =  50  cm,  etc…    Make  sure  all  columns  and  rows  are  named  with  appropriate  units.  

Discussion:  The  10%  error  (my  guess)  is  primarily  the  result  of  the  variable  landing  points  of  the  marbles  due  to  starting  the  marble  at  different  height,    possible  movement  of  construction  paper,  the  corn-­‐starch  coating  slowing  the  marble.        

Conclusion:  This  experiment  verified  the  applicability  of  the  equations  of  motion  with  the  experimental  errors,  and  thereby  the  initial  hypothesis  has  been  confirmed          

References:    Cite  the  sources  used  for  the  Background  info  or  other                

Abstract The goal of this experiment was to determine the effect of mass and length on the period of oscillation of a simple pendulum. Using a photogate to measure the period, we varied the pendulum mass for a fixed length, and varied the pendulum length for a fixed mass. The results of this experiment are in close agreement with theory: mass had no measurable effect on the period of our pendulum, while the data for period vs. length is well-described by a power-law relationship close to the theoretical square-root dependence.

Introduction Theoretically, the period of a pendulum is independent of its mass, and depends on length according to the power-law relationship

∝T L

where T is the period of oscillation and L is the length. This result can be determined using a dimensional-analysis approach. The independence of mass is a result of the fact that all objects are accelerated towards the center of the earth with the same acceleration of gravity.

Procedure/Data Analysis A mass was attached to a string and suspended from a ring stand. A piece of paper with an angle drawn on it was used to start the pendulum from the same initial angle for each measurement. The pendulum swung through a photogate timer connected to a PASCO interface box, and data was collected using Science Workshop

All data collection started a few seconds after the pendulum's release, allowing the pendulum to settle down into a regular motion. The pendulum was allowed to swing through 4-6 oscillations. Scientific Workshop was used to calculate the average period of oscillation. These periods were then recorded for a set of different masses for the same length of string, and then for a set of different string lengths for the same mass.

Mass variations For this part of the experiment the length was fixed at 0.6 m.

mass (g) T (s) 50 1.56 100 1.59 250 1.56 375 1.57 400 1.56

simple pendulum exp

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The data was copied into Excel and graphed. Theory predicts that there should be no effect of mass on period.

Length Variation For this part of the experiment the mass was fixed at 100 g.

The data was copied into Excel and graphed. A power law was fit through the data.

L (m) T(s) 0.15 0.83 0.30 1.15 0.45 1.36 0.50 1.48 0.55 1.51 0.60 1.57 0.90 1.93

Mass Effect on Pendulum Period (L = 0.6 m)

0.00

0.50

1.00

1.50

2.00

2.50

0 100 200 300 400 500

m (g)

T (s

)

Length Effect on Pendulum Period (M = 100 g)

T = 2.2144 L0.4918

R2 = 0.9455

0.00

0.50

1.00

1.50

2.00

2.50

0.00 0.20 0.40 0.60 0.80 1.00

L (m)

T (s

)

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Conclusions Our period vs. mass measurements showed almost no noticeable variation with mass. Theory predicts no variation at all. Our variations may have been due to the fact that, as the masses were varied, the length of the pendulum may have changed as the string was unattached and attached. Another possibility for the variation was that the photocell might have reacted differently with the different shapes and materials. The variations appear to be due to a lack of precision because they appear random.

The period vs. length measurements were well described by a power law. Using Excel to fit the data, the best power-law fit is:

0.45952.033T L=

Theoretically, the result should be 0.52.01T L=

Our exponent is quite close to the theoretical value of 0.5, and therefore our data is consistent with the pendulum’s period depending on the square-root of length. Discrepancies in the power and in the leading term are most likely due to precision errors in the measurement of pendulum length. These errors occur as we try to measure the length of the pendulum from where the string connects to the ringstand to the middle of the mass. Determining the middle is not very exact, and hence the L measurements suffer from a lack of precision..

Note also that the angle, though controlled, was also not precise. This experiment was not designed to look for starting angle effects on a pendulum's period. Therefore we cannot rule out starting angle effects leading to discrepancies between our results and the theoretical prediction.

g

Friction Experiment

1. Introduction

Pushing a heavy box that is lying on the ground can be quite difficult for even a large muscular adult. However, if the same box is placed on a wheeled dolly instead, a small child may have little difficulty moving the box. How is this possible? Does the dolly somehow make the box any lighter? Friction is at the root of this issue. If you try to move the box while it is on the ground, you will encounter a significant amount of friction, which must be overcome. Placing the box on the dolly reduces the friction present in the system and makes it much easier to move the box. It may sound as though friction is something that makes life overly difficult and that you would be better off without it. The truth is very much the opposite, however. Cars use friction both to enable them to move and to brake while in motion. Have you ever tried to walk on ice? Even on ice there is a small amount of friction. Without friction, walking would be impossible. In this experiment we will classify the various forms of friction that are encountered in everyday life.

2. Background

2.1. Force

What is commonly referred to as friction is actually a frictional force. Therefore, to properly define friction, we must first introduce the concepts of force and velocity. Force and velocity are vector quantities; they have both a magnitude and direction. Velocity can be thought of as the speed in a specified direction, with which a given object is traveling. Force is the mechanism through which the object accelerates, or undergoes a change in its velocity. A body that is at rest is one in which the resultant force, the sum of all forces acting on the body, is zero. Such an object is said to be in equilibrium. Although many different forms of forces exist, there are two broad classifications that are commonly used to categorize forces. First, we distinguish between conservative and dissipative forces. Conservative forces act upon an object in a manner such that the total energy of the object is conserved. Dissipative forces are ones that reduce the total energy of the object upon which they act. Friction is a dissipative force. A different way to categorize forces is to distinguish between contact and field forces. Contact forces, as the name implies, rely upon physical contact to act on a body. Field forces require no physical contact to act on a body. Friction is a contact force. Gravitational and magnetic forces are common examples of field forces. See [8] for more details. We will now define and discuss a few types of forces that are of direct importance to our discussion on friction.

Gravitational force

Gravity is an attractive force that exists between two masses. The magnitude of the force depends on the proximity of the masses involved, as well as the magnitude of the masses. At large distances, and/or if the masses of interest are of small quantity, the effects of gravity can be ignored. That is why astronauts in outer space float freely, and astronauts on the moon bounce along as they do. We approximate the gravitational force that acts on an object on Earth as

F = mg

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where m is the mass of the object and g is the free-fall acceleration. This value is only an approximation because the magnitude of g depends on the altitude of the object as well as the latitude of the location. However, because the spatial differences in acceleration due to gravity are so small, the approximation is well justified. The magnitude of the gravitational force exerted on an object is referred to as its weight. See [8] for more details.

Normal force

The normal force is a contact force that acts in the normal direction, i.e., perpendicular to, the contact surface between two objects. For an object subjected to no forces other than gravity, the magnitude of the normal force is equal to the component of the gravitational force that acts normal to the surface, but it acts in the opposite direction. See [2, 8] for more details. Figure 1 depicts a free-body diagram of a mass, ,m lying on a horizontal surface, where the normal force, ,N is equal in magnitude to the gravitational force .mg

Figure 1: Free-body diagram of a mass lying on a horizontal surface

2.2. Friction

There are different forms of frictional forces that occur. When friction acts on an object that is at rest, we refer to the frictional force as static friction. An object that is in motion is subject to kinetic, or dynamic, friction. Friction is a resistive force, one that damps out motion in dynamic systems and prevents movement in static systems. Friction occurs because an object interacts with either the surface it lays upon, the medium it is contained in, or both. Only in a complete vacuum can a system be free of friction.

Static friction

When you want to push a heavy object, static friction is the force that you must overcome in order to get it moving. The magnitude of the static frictional force, ,sf satisfies

Nf ss µ≤

where sµ is the coefficient of static friction, a dimensionless constant that depends on the object and the surface it is laying upon. From this equation it is clear that the maximum force of static friction, ,max,sf that can be exerted on an object by a surface is

m

mg

N

.max, Nf ss µ=

Once the applied force exceeds this threshold the object will begin to move. A common example of a static friction force is that of a stationary mass on an incline. Figure 2 depicts the free-body diagram of this case.

Figure 2: Free-body diagram of a mass on an incline

If the angle at which the mass begins to slide is known, we can determine sµ by decomposing the forces into the Cartesian coordinates, ,, yx as given in Figure 2. Since we are interested in the instant at which movement begins, we are dealing with an object in equilibrium. Thus, the resultant force in both the x and y directions must be zero. Analysis of the forces in the x direction yields

.sinθmgf s = (1)

Following a similar procedure for the y direction yields

.cosθmgN = (2)

However, we know that at the instant that the mass begins to move

.max, Nff sss µ== (3)

Thus, by substituting (2) into (3) and equating the result with (1) we can solve for sµ as follows

.tansincos

θµθθµ

=⇒

==

s

ss mgmgf

Kinetic friction

Once the force applied on a mass exceeds max,sf and the mass begins to move, a kinetic friction

force, ,kf exists. Kinetic friction coefficients are generally less than static friction coefficients, which is the reason that it is much easier to keep a heavy object in motion than it is to start it in motion [4]. The magnitude of the kinetic frictional force is given as follows

Nf kk µ=

θ

y

xsf

mgN

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where kµ is the coefficient of kinetic friction and is approximately constant. Figure 3 is a plot of frictional force versus applied force [1, 8]. The slight fluctuations in the kinetic region of the plot are due to the slight dependency of kµ on the speed of the object [8].

Figure 3: Plot of frictional force versus applied force

Table 1 lists the coefficients of static and kinetic friction for various object/surface combinations [5, 8]. The coefficient of static friction is less than that of kinetic friction for all cases in the table except Teflon on Teflon.

Table 1: Coefficients of static and kinetic friction

Materials Coefficients of Static

Friction )( sµ Coefficients of Kinetic

Friction )( kµ Steel on Steel 0.74 0.57 Aluminum on Steel 0.61 0.47 Copper on Steel 0.53 0.36 Rubber on Concrete 1.0 0.8 Wood on Wood 0.25-0.5 0.2 Glass on Glass 0.94 0.4 Waxed wood on Wet snow 0.14 0.1 Waxed wood on Dry snow - 0.04 Metal on Metal (lubricated) 0.15 0.06 Ice on Ice 0.1 0.03 Teflon on Teflon 0.04 0.04 Synovial joints in humans 0.01 0.003

Rolling friction

Rolling friction is a special case of kinetic friction. When we say that a wheel rolls without slipping, we are implying that the wheel is subjected to rolling friction. The motion of a wheel under rolling conditions is not affected by kinetic friction. There are no dissipative forces acting on the wheel subject to rolling friction, under ideal conditions, because the wheel does not slip at the point of contact. As a result, rolling friction does not impede on the motion of the wheel. However, once rolling ceases, kinetic friction is the only frictional force acting on the wheel.

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connected to the base through a hinge, is rotated by a DC motor. The DC motor is driven by a regulated 12 VDC external power supply. An H-Bridge is used to control the rotational direction of the DC motor. An InfraRed (IR) transmitter and receiver pair is used to detect movement of the mass. The pair has been mounted on the opposing sidewalls of the plate facing each other. The IR transmitter continuously sends an infrared pulse towards the receiver. When the mass starts to slide, it interrupts the infrared beam. To ensure accuracy, the experiment must be conducted so that the first instant of the mass’s movement is detected. To achieve this accuracy, a stopper has been placed at the free end of the plate such that by placing the mass snug against it the IR beam will be tripped the instant the mass begins to slide. Once the IR receiver has been tripped, the BS2 shuts the motor down. The angle at which the mass begins to slide is measured by a rotary potentiometer connected to the hinge on which the plate turns. A constant 5VDC voltage is applied across the full resistance of the potentiometer. The wiper rotates along with the plate, and the voltage between the wiper and one end of the potentiometer is measured. Since the BS2 is a digital device and the quantity being measured is an analog voltage, an Analog to Digital Converter (ADC) IC is utilized. The quantization number, ,Q of an ADC depends on the number of bits the ADC has and on the span (range of values) that is being digitized. Q for an n -bit ADC is given by

nspanV

Q2

=

where spanV is the voltage span to be digitized. This test bed uses an ADC0831 from National Semiconductor, an 8-bit, successive approximation converter that is easily interfaced with the BS2 using the PBasic shiftin command. See [6, 7] for more details. As an 8-bit device it can convert an analog signal into 256 discrete levels. To maximize the sensitivity of the ADC the offset and span voltages were provided to the ADC by adjusting two potentiometers to provide the proper voltages. A stopper has been installed at the foot of the plate to prevent the mass from falling down. Circuit schematics for the static friction coefficient can be found in Appendix A.

Figure 5 is an overhead view of the test bed. The DC motor can be seen in the lower left hand corner of the figure. The potentiometer can be seen in the upper left hand corner. The stopper that prevents the mass from falling of the plate can be seen on the left hand side of the figure. The electrical circuitry used by the test bed can be seen on the right hand side of the figure.

Figure 5: Overhead view of the static friction coefficient test bed

Potentiometer

Stopper

DC motor

Circuitry

H-Bridge

ADC with span and offset

Experiment : Momentum

Physics is often concerned with what are called “conserved” quantities. Mass and energy are twoexamples of quantities that must remain conserved for a closed system. Conservation of a quantity is aclue to a physicist that there is some underlying principle to be discovered. Perhaps the oldest and mostfamous conservation principle is the conservation of momentum. This is embodied in Newton’s First Law,written in 1687. It states that an object in motion will remain in motion unless acted upon by a net force.Conservation of momentum will be studied through one dimensional collisions.

One Dimensional Collisions

The concept of momentum is fundamental to an understanding of the motion and dynamics of anobject. The momentum of an object is defined to be

−→p = m−→v (1)

For multiple objects in a system, the total momentum is a vector sum of the individual momenta. As aconsequence of Newton’s second law

−−→Fext =

d−→pdt

(2)

For a closed system, the total momentum cannot change unless acted upon by an outside force. Thisconservation of momentum is a powerful tool for physicists to analyze the behaviors of systems of particles.The simplest application of this concept is in the one-dimensional collision between two particles. Thereare two special kinds of collisions which are particularly easy to analyze: the perfectly elastic and perfectlyinelastic collisions. While both of these processes conserve momentum, in the perfectly elastic collision thetotal kinetic energy, KE, is also conserved. Examples of perfectly inelastic collisions include objects whichcollide and stick together and objects which break apart due to internal forces.

In the analysis of a perfectly elastic one-dimensional collision, consider two objects with masses m1

and m2 and initial velocities v1 and v2. After the collision, the objects will have new velocities v′1 and v′2,where all velocities are assumed to be in the positive direction. Conservation of momentum demands thatthe total momentum must be the same before and after the collision. This can be stated as:

m1v1 + m2v2 = m1v′1 + m2v

′2 (3)

Since the kinetic energy is also conserved in this kind of collision, we have:

1

2m1v

21 +

1

2m2v

22 =

1

2m1v

′21 +

1

2m2v

′22 (4)

If we know the masses and the initial velocities, it is possible to solve for the final velocities of the twoobjects. After a number of algebraic manipulations, the solutions are:

v′1 =m1 −m2

m1 + m2v1 +

2m2

m1 + m2v2 (5)

v′2 =2m1

m1 + m2v1 +

m2 −m1

m1 + m2v2. (6)

Again, all velocities are presumed to be along the positive direction. If a velocity is negative, it is thendirected along the negative direction.

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The perfectly inelastic (“sticky”) collision is somewhat easier to analyze as only Equation 3 can beused. The energy equation used in the analysis depends on which case is being studied. If the collisionstarts with a single object which breaks into two, then we have v1 = v2. If there are initially two objectswhich end up stuck together, then we have v′1 = v′2. The analysis is then straightforward.

Most collisions are neither perfectly elastic nor perfectly inelastic but partially elastic. This means thata certain fraction of the kinetic energy is lost to the system but the objects do not stick together. In thiscase, it is valuable to define a quantity called the coefficient of restitution

e =v′1 − v′2v2 − v1

(7)

For a perfectly elastic collision, e = 1 and for a perfectly inelastic collision (starting with two bodiesand ending with one), e = 0. For most real collisions 0 < e < 1.

Experimental Objectives

In the laboratory you have air track, two gliders, two photogates, a scale, and additional masses thatcan be placed on the gliders. The glider carts have velcro on the ends to create a “sticky” collision orrubber bands for an elastic collision. The photogates are connected to a computer data acquisition systemand velocity data can be collected using the Data Studio software (ask your TA about using this softwarein your experiment and setting up the two photogates). Remember, velocity is a vector, so you must assignit a direction - the Data Studio software cannot do this. Using this equipment:

• Devise an experimental procedure to observe and verify linear momentum and energy conservationlaws. Repeat your experiment for different masses and different velocities of the glider (includingzero initial velocity for one of the gliders). Do five different collisions and repeat each one to getconsistent data.

• Devise an experiment to study “perfectly” inelastic collisions. Verify whether or not momentum andenergy are conserved in this type of collision.

A full lab report is not necessary for this lab. Answer the questions on the following page and turn itin with your signed datasheet.

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