Physics 101

Prof. Ekey

Chapter 8 + 13

Dynamics II: Motion in a plane 2D Fnet, centripetal force, bucket of water, gravity “In this chapter, you will learn to solve problems about motion in two dimensions.”

“In this chapter, you will understand the motion of satellites and planets.”

Please take a look back at rotations (chapter 4).

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ax =Fnet( ) xm

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ay =Fnet( )ym

(a) (b) (c) (d) Path not shown

Dynamics in 2D. Use net forces to find x and y acceleration. Then use kinematic equations to find pos, vel or time. Remember projectile motion? That was fun. Question:

A hockey puck slides to the left at 1.0 m/s on a 2D horizontal (level) surface. A rocket on the puck is aimed in the horizontal direction, and is fired (X). Which of the following shows the path of the puck post-fire? Is the speed constant or changing as the rocket fires?

X X X

Rocket Man

A 500 g model rocket is on a cart that is rolling to the left at a speed of 3.0 m/s. The rocket engine, when it is fired exerts an 8.0 N thrust on the rocket. Your goal is to have the rocket pass through a small horizontal hoop that is 20 m above the launch point. At what horizontal distance should you launch?

Use NII to find the acceleration in the y-direction. Net force in x? Use y-component knowledge to find the flight time.

Then use x-component knowledge to find x1.

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vt =dsdt

= r dθdt

vt = rω

Chapter 4 - review Angular Displacement, ∆q Units: radians Angle turned through

Angular Speed (w) Units: rad/sec Average Angular Speed Instantaneous Angular Speed

Tangential (linear) and angular speeds.

“A particle moving in a circle has an instantaneous velocity tangential to its circular path.”

Tangential speed, v, related to w

Δθ = θ − θo

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ωavg =ΔθΔt

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ω = limΔt→0

ΔθΔt

=dθdt

1 rad = 57.3º

only care about final and initial location 1 rev = 2π rad

Uniform circular motion – movement at constant speed in a circular path

r v

direction of rotation

Direction of w?

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v =2πrT

Period, T - Time it takes for one revolution. Units: seconds

Frequency, f – revolutions per unit time. Units: Hertz (Hz)= 1/sec

Tangential speed for uniform circular motion: (distance traveled in 1 revolution per period)

Angular Acceleration, a Units: rad/s2

Average angular acceleration (time rate of change of angular velocity)

Tangential acceleration, at With an angular acceleration, tangential speed changes magnitude.

f = 1

T

ω =

2πT

= 2π f

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αavg =ΔωΔt

at = rα

Period is inversely related to frequency

Angular speed in terms of period and frequency

Analogous to avg. linear acceleration Points in the same or opposite direction as angular velocity

Vector

magnitude of tangential acceleration

at constantly changes direction

Vector

Centripetal acceleration, ac – “center-seeking” acceleration

Centripetal acceleration is perpendicular to the instantaneous velocity & always towards the center of motion.

magnitude of centripetal acceleration Also called “radial” acceleration, ar

ac =

v2

r=

(rω )2

r= rω 2

Uniform circular motion – movement at constant speed in a circular path

Tangential speed remains constant Direction of velocity changes

No acceleration in tangential direction

If there were the magnitude of the tangential speed would change.

Question You are a passenger in a car and are not wearing a seatbelt (shame on you). Without increasing or decreasing your cars speed, the car makes a sharp left turn, and you find yourself colliding with the right-hand (passenger) door. Which is the correct analysis of the situation? (a) Before and after the collision, there is a rightward force pushing you into the door.

(b) Starting at the time of collision, the door exerts a leftward force on you.

(c) both of the above. (d) neither of the above.

Centripetal Force (plug ac into NII law for anet)

Keeps the object moving in a circle (causes centripetal accelerations)

Not a new force, supplied by tension, force of gravity, friction… etc.

A 0.50 kg ball is attached to a string with a radius of 1.0 m. You spin the ball above your head with a speed of 5.0 m/s. Calculate the centripetal force. What causes the centripetal force? What is the net force on the ball? If the string breaks when a tension greater than 50 N is applied, how fast do you have to spin the ball to cause the string to break?

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Fc = mac = m v2

r= mrω2

points radially inward toward center of motion (same as centripetal acceleration)

Say “NO” to centrifugal force

Broken string… If the string breaks as shown, what direction does the ball travel? Question: A ball on a string spins with a constant tangential speed. If you halve the radius of the string while halving the tangential speed, the centripetal force will be… (a) Twice the original centripetal force (b) Half the original centripetal force (c) The same as the original centripetal force (d) One quarter the original centripetal force

Reasoning?

Both point towards center of circle

“good choice of coordinate system for uniform circular motion” force/acceleration only have a radial component

3D coordinate system for circular motion (rtz)

r – axis (radial axis): Points from the particle toward the center of the circle. t – axis (tangential axis) : Tangent to the circle, pointing in the ccw direction. z – axis – perpendicular to the plane of motion (follows RHR)

In uniform circular motion object only has tangential velocity radial acceleration and centripetal force

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vr = 0vt = rωvz = 0

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ar =v2

r= rω2

at = 0az = 0

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Fnet( )r = Fr∑ = mar = m v2

r= mrω2

Fnet( )t = Fr∑ = 0

Fnet( )z = Fr∑ = 0

The tires are the things on your car that make contact w/the road A 1500 kg car take a 50-m radius unbanked curve at 15 m/s, barely keeping the car in the turn (almost slips). What is the magnitude & direction of the centripetal force on the car? What is the magnitude & direction of the force of friction on the car? What type of friction is this? Calculate the normal force on the car. What is the coefficient of friction between the tires and the road?

Question A rider in a "barrel of fun" finds herself stuck with her back to the wall. Which diagram correctly shows the forces acting on her?

Identify all forces acting on her

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vcritical = rg and ωcritical = vcritical r = g r

Spinning a bucket of water vertically Whirl a bucket with a speed vt at a radius r = 1.0 m. Look at forces on the bucket at top of motion.

Normal (support) force from pail + Force due to gravity.

Inc. (dec.) vt & the pail exerts a larger (smaller) force on the water.

What’s the minimum speed of the pail to keep the H2O in the pail? Critical Speed – Speed where gravity alone is sufficient to cause circular motion at the top. (Normal force is zero. Bucket +water)

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Fnet = Fr∑ = n+mg = ma = m vt2

r

Difficulty: No spilling

n

Fg

vt

Accel. towards center

Ball spun in vertical plane A ball is spun in a vertical plane with constant velocity. Where it the string most likely to break? Or, where is the greatest tension in the string? (a) At the top. (b) At the bottom. (c) On the sides. (d) It doesn’t matter. Draw a force diagram or two If the ball’s mass is 1.0 kg, and the string is 1.0 m long and it spins with a constant speed of 10 m/s. Calculate the tension at the top and bottom.

v

Loop-da-Loop

If the car is right-side up at the top, what changes?

Roller coaster car in a vertical loop.

Draw a force diagram for the top and bottom and bottom of the loop Where is the speed the greatest? Least? Is this uniform circular motion? How is the speed changing at the top and bottom? atangential at top/bottom?

If you are traveling at twice the critical speed at the top of the loop, what is the ratio of the normal force to the gravitational force? Assume r=20m

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F1 on 2 = F2 on 1 =Gm1m2

r 2

Chapter 13 (abridged) Force of gravitational attraction between two masses. Calculate the force of gravity on a 1 kg mass at the surface of the earth. Calculate the acceleration due to gravity at the surface? What happens if you go to double the radius? What happens to the force of gravity

Use in Chap13 HW questions, not Ch8 G = 6.67 x 10-11 Nm2 / kg2 gravitational constant

me = 5.98 x 1024 kg re =6.37 x 106 m

Measured period of the moon orbiting the earth = 27.3 days = 2.36x106 s

mE=5.98 x 1024 kg mm=7.36 x 1022 kg rEtom=3.84 x 108 m

Some would say the earth is our moon. Orbit = Close trajectory around an object (planet) Which way does gravity point? Straight down? How about towards the center of the earth?

Orbiting object is in “free-fall” (Can’t assume g=9.8 m/s2) What’s the value of “g” from the earth at the orbit of the moon? Setup sum of all forces, solve for “a”. What is the tangential speed of the moon orbiting the earth? What’s the period of the moon orbiting the earth?

Questions The figure shows a binary star system. The mass of star 2 is twice the mass of star 1. The magnitude of the force of planet 1 on 2 is ______ as the force of planet 2 on 1? (a) four times as big (b) twice as big (c) the same (d) half as big (e) one-quarter as big

Three stars are aligned in a row. The net force on the star of mass 2M is

(a) To the left. (b) To the right. (c) Zero. (d) Not enough information to answer.