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Unit 2CStatics

Static Equilibrium

Translational Forces

Torque

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Dynamics vs Statics

Dynamics: is the study of forces and motion.

We study why objects move.

Statics: is the study of forces and NO motion.

We study why objects DO NOT move.

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Recall Newton’s First Law

All objects remain at rest, or continue to move at a constant velocity unless acted upon by an external unbalanced force.

Thus, an object will stay at rest if all of the forces acting on the object balance each other.

We say that the object is in a state of STATIC EQUILIBRIUM if it is at rest.

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An object moving with a constant velocity is in a state of DYNAMIC EQUILIBRIUM.

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In Static Equilibrium

We must consider:

Translation Forces

Forces that make objects move from one place to another.

Rotational Forces

Forces that make an object rotate about a point.

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Translation Forces

Are the following in a state of static equilibrium?

Note: The first two are 1-dimensional problems (2204), and the last is a 2-dimensional problem (3204)

FBD

While force components are needed to solve the problem, they are not officially part of the free body diagram.

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0xF 0yF

Horizontal Forces Vertical Forces

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Translation Forces

Activity

Draw a free body diagram.

What is the mass of the object?

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Find the tensions in the ropes below.

0xF

Horizontal Forces Vertical Forces

60.0 kg

T2 T1

T2

T1

1 2 0x xT T

1 2sin42 sin26o oT T

1 2x xT T

0yF

1 2 0y y gT T F

1 2y y gT T F

1 2cos42 cos26o oT T mg

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sin26

sin42

o

o

TT

22 cos42sin

cos226

sin6

42o

o

oo m

TT g

2 cos42 csin26

os26sin42

o oo

o gT m

1 sin42oT

2 sin26oT

2 cos26oT

1 cos42oT

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2

cos4sin26

sin422 cos26

oo

oo

mgT

2 sin26

sin

60 9.8

cos42 cos242

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Nkg

o oo

o

kgT

2 424T N

1 278T N

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sin26

sin42

o

o

TT

sin26

sin42

424 o

o

N

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OR

60.0 kg

T2 T1

T1

T2

mg

Why do the vectors connect together?

The net force is zero

We can find the tensions using the Law of Sines

sin sin sinA B Ca b c

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Find T1

60.0 kg

T2 T1

T1

T2

mg

180 42 26o o o

We need to find the missing angle first.

1

sin26 sin112

T mg

112o

1 sin112 sin26T mg

1

60.0 9.8 sin26

sin112

Nkgkg

T

1 278T N

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Find T2

60.0 kg

T2 T1

T1

T2

mg

2

sin42 sin112

T mg

2 sin112 sin42T mg

2

60.0 9.8 sin42

sin112

Nkgkg

T

2 424T N

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Find the mass of the snowflake

45.0

o

10.0 N

30.0

o

T1

T1

10.0 N

mg

We need to find the missing angle first.

180 45 30o o o 105o

sin45 sin105

10.0N mg

sin45 10.0 sin105mg N

10.0 sin105

9.8 sin45Nkg

Nm

1.39m kg

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Tension Worksheet

1. A guy wire attached to the top of a gate post is

anchored to the bottom of the next post, as illustrated. Assuming that the gate post does not move, what are the horizontal and vertical components of the force exerted on the top of the post by the guy wire, if the tension in the wire is 200 N?

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2. A clothesline is attached to high poles 10.0 m apart. A pulley, allowed to roll freely on the line, has a 25 kg mass hanging from it. Find the tension in each half of the clothes -line if the sag at the center is 0.40m as shown.

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3. A sphere of mass 4.0 x 10 -4 kg is suspended from a string. A steady horizontal breeze pushes the sphere so that the string makes an angle of 60o with the vertical when at rest. Find the magnitude of the push of the wind.

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4. For the 850 g picture below, find the tension in the string if the string makes angle of 82o at the nail.

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5. Find the tension in string 1 and 2 below

1 2

37 53o o

2.0 kg

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6. Find the mass of the block

50o o32

15N

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7. What is the tension in T1?

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Torques(Rotational Forces)

Give me a lever long enough and a fulcrum on which to place it, and I shall move the world.

Archimedes

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Watch what happens when you don’t study Torque

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Boss man, you ain't gonna believe this!I left my hook hanging in the water while we were at lunch and a BIG fish almost pulled my rig into the water.

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Centre of Mass(Gravity Spot)

The centre of mass is a single point in a body at which its entire mass is considered to be concentrated.

An object can balance on a point only if its center of mass is directly above the point.

Alternatively, if you hang an object from a string, the object's center of mass will be directly below the string.

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It is usually located near the more massive part of an object.

The centre of mass is also called the centre of gravity, which is the point at which the force of gravity acts. The force of gravity is equal on both sides of the object’s centre of gravity.

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The center of mass is an important point on an aircraft, as it defines the amount of mass forward or behind the center of gravity that needs to be moved in order to pitch the plane up or down without applying any external forces

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Location of Centre of Mass

For uniformly, regular shaped objects the centre of mass is at the geometric centre. For example the centre of mass of a ball

(sphere) is located at the centre of the ball.

For non-uniform shaped objects, the centre of mass is located at its balancepoint in a gravitational field.

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Example:Find the centre of mass of the following:

1. A uniform 10 m log?

2. A uniform 15 m extension ladder?

3. A broom?

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4. A Triangle

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6.A boomerang? Note: In the case of a boomerang the centre of mass is not located within the actual boomerang!

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A male versus A femaleThe chair experiment

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A person's center of mass is slightly below his/her belly button, which is nearly the geometric center of a person.

Males and females have different centers of mass Females' centers of mass are lower than those of

males. The average ratio of center of mass to height in

females is approximately 0.543 and the average ratio of center of mass to height in males is approximately 0.560.

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Torque

When a force is not directed through an object’s centre of mass, the force will cause the object to rotate.

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Torque

Torque: The rotational force caused by a force acting at a distance from a pivot point (fulcrum).

Examples: Opening a pop bottle

Tightening a screw

Using brakes in a car

Opening a door

Opening a paint can with a lever

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Equation for Torque

where

is Torque

r is the distance a Force is applied

from a pivot point (aka point of rotation)

is the component of the force that acts perpendicular to the surface.

What are the units for Torque?

F r

F

Nm

If the force acting at an angle, , to the rigid arm (lever) then torque can be calculated using the formula:

Note: Cosine can be used if the angle is given from the perpendicular of the lever

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sinr F

Pivot

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Is Torque a vector quantity?

Yes! The direction is:

clockwise or counterclockwise

Clockwise is takento be positivein the text.

Counterclockwise is takento be positive in mathematics and inengineering.

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Examples of Torque

1.Find the torque supplied to a 75 cm wide door by the following forces applied at the edge of the door:

A) 20N

B) 30 N at 30o to the perpendicular to the door

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2.Suppose a bolt requires 65 Nm to be properly tightened. What force is required if the wrench is 15 cm long?

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Not tested on public, but useful info in real life!!

3.In the old British system (and in the US) torque is measured in ft-lbs. How many Nm is one ft-lb?

One foot is 0.305 m.

One kilogram is 2.2 pounds

A) 80 ft-lb is ____ Nm B) 15 Nm = ______ ft-lb

4. A plumber applies a force of 14 N to a wrench at an angle of 60.0o as shown. What is the length, r, if the plumber produces a torque of 4.6 Nm?

(A) 0.33 m

(B) 0.38 m

(C) 0.66 m

(D) 0.72 m

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Static EquilibriumConsists of 2 Parts

First Part

Translation Forces

where is the sum of ALL of the

forces acting on an object through its centre of mass

1 2 3 ...net nF F F F F 1

n

ii

F

0

netF external

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Static EquilbriumConsists of 2 Parts

Second Part

Torques (or Moments of Force)

where is the sum of ALL of the

(or moments of force) acting on an object.

NOTE: These forces are NOT acting through the object’s centre of mass. WHY NOT?

There is NO rotation force (torque) if a force acts through an object’s centre of mass

1 2 3 ...net n 1

n

ii

0

net torques

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Static Problems Torque(we did translational forces in unit 2)

Seesaw Problems

Cantilever Problems

Crane Problems

(or Strut and cable )

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Torque QuestionsSeesaw problems:

1. Pat and Tyler are playing on a 4 m long seesaw that is supported at the centre. If Pat has a mass of 30 kg and sits at one end of the seesaw, where should Tyler (mass = 35 kg)

sit so that the seesaw balances?

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2.A playground seesaw with a total length of 5.0 m and a mass of 30.0 kg is pivoted at its center. A 20.0 kg child sits on one end of the seesaw.

a) Where should a person push with a force of 220 N in order to hold the seesaw level?

b) Where should a 40.0 kg child sit to balance the seesaw?

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3. Consider the diagram below :A load weighing 60 N is to be supported by a force F applied at the end of a 5.0 kg, uniform, lever as shown. What Force is necessary if the fulcrum is placed at position A?

A

1.2m

40cm

60 N

F

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4. What force is needed to balance the 10.0 kg mass if:

2.0 m

1.0 m

10.0 kg

F

A) the seesaw is massless. B) the seesaw has a mass of 42 kg

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5. In order to hang a mass of M1=30.0kg from the horizontal flat roof of a building, a plank of length 2.4 m is placed on the roof. A rock of mass M2= 15.0 kg is placed on one end. How far can the end of the plank reach over the building without tipping over?

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Cantilever Problems1. A 1.5 x 10 3 kg car is crossing a 120 m long flat bridge

which is supported at both ends. When the car is 32 m from one end, what force must each end support be able to provide?

A B

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2. A 4.0 m diving board is supported by two blocks, one at the end and the second 1.0 m from the end. A 60.0 kg person stands at the end of the board. Find the forces given to the board by the two supports if:

A)The board is a Canadian Tire special - no mass.

A B

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2. A 4.0 m diving board is supported by two blocks, one at the end and the second 1.0 m from the end. A 60.0 kg person stands at the end of the board. Find the forces given to the board by the two supports if:

B) The mass of the board is 40.0 kg.

A B

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3.In the diagram below, a 10.0 m uniform horizontal beam, weighing 1.00 × 102 N is supported by a rope at each end. If a 4.00 × 102 N box is positioned 2.0 m from the left end of the beam, what is the tension in each of the support ropes (T1 andT2)? (June 2005 - Public)

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Crane Problems (or Suspended load from a strut and cable)

1. Determine the tension in the cable and the compression force in the boom to support the 1.0 x 10 2 kg object. The angle between the boom and the supporting cable is 37° .

Popular PublicQuestions

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2. A crane is used to lift a uniform 24 m long pipe with a mass of 730 kg as shown in the diagram below.

What minimum tension is required by the cable to lift the end of the pipe off the ground?

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3. A uniform rod of length L and mass of 4.0 kg is hinged at the left end. A 25.0 kg sign is suspended from the right end. A guy

wire is connected to the end of the rod and fastened to a wall.

A) Draw a free body diagram for the rod .

B) Determine the tension in the guy wire.

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June 2004 Public

(i) Sketch the free body diagram for the rod in the diagram above. Label all forces. (2 marks)

(ii) If the mass of the block is 5.0 kg and the rod is uniform with a mass of 0.40 kg, what is the magnitude of the tension in the wire? (3 marks)

Answer

5. A uniform 15 kg beam of length 4.0 m is supported against a wall as shown. A 25 kg object is suspended 3.0 m from the hinge P.

A) Draw a FBD

B)What is the tension in the support cable?

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