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General Physics (PHY 2130)

Lecture VII

- Rotational dynamics
- torque
- moment of inertia
- angular momentum
- conservation of angular momentum
- rotational kinetic energy
- Exam II review

http//www.physics.wayne.edu/apetrov/PHY2130/

Lightning Review

- Last lecture
- Rotations
- angular displacement, velocity (rate of change of

angular displacement), acceleration (rate of

change of angular velocity) - motion with constant angular acceleration
- Gravity laws

Review Problem 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

Sample Review Problem

An engineer wishes to design a curved exit ramp

for a toll road in such a way that a car will not

have to rely on friction to round the curve

without skidding. She does so by banking the road

in such a way that the force causing the

centripetal acceleration will be supplied by the

component of the normal force toward the center

of the circular path. Find the angle at which the

curve should be banked if a typical car rounds it

at a 50.0-m radius and a speed of 13.4 m/s.

- Rotational Equilibrium
- and
- Rotational Dynamics

Torque

- Consider force required to open door. Is it

easier to open the door by pushing/pulling away

from hinge or close to hinge

close to hinge

away from hinge

Farther from from hinge, larger rotational effect!

Physics concept torque

Torque

- Torque, , is the tendency of a force to rotate

an object about some axis - is the torque
- d is the lever arm (or moment arm)
- F is the force

Door example

Lever Arm

- The lever arm, d, is the shortest (perpendicular)

distance from the axis of rotation to a line

drawn along the the direction of the force - d L sin F
- It is not necessarily the distance between the

axis of rotation and point where the force is

applied

Direction of Torque

- Torque is a vector quantity
- The direction is perpendicular to the plane

determined by the lever arm and the force - Direction and sign
- If the turning tendency of the force is

counterclockwise, the torque will be positive - If the turning tendency is clockwise, the torque

will be negative

Direction of torque out of page

An Alternative Look at Torque

- The force could also be resolved into its x- and

y-components - The x-component, F cos F, produces 0 torque
- The y-component, F sin F, produces a non-zero

torque

L

F is the force

L is the distance along the object

F is the angle between force and object

Lets watch the movie!

ConcepTest 1

You are trying to open a door that is stuck by

pulling on the doorknob in a direction

perpendicular to the door. If you instead tie a

rope to the doorknob and then pull with the same

force, is the torque you exert increased Will it

be easier to open the door 1. No 2. Yes

Please fill your answer as question 29 of

General Purpose Answer Sheet

ConcepTest 1

You are trying to open a door that is stuck by

pulling on the doorknob in a direction

perpendicular to the door. If you instead tie a

rope to the doorknob and then pull with the same

force, is the torque you exert increased Will it

be easier to open the door 1. No 2. Yes

Convince your neighbor!

Please fill your answer as question 30 of

General Purpose Answer Sheet

ConcepTest 1

You are trying to open a door that is stuck by

pulling on the doorknob in a direction

perpendicular to the door. If you instead tie a

rope to the doorknob and then pull with the same

force, is the torque you exert increased Will it

be easier to open the door 1. No 2. Yes

ConcepTest 2

You are using a wrench and trying to loosen a

rusty nut. Which of the arrangements shown is

most effective in loosening the nut List in

order of descending efficiency the following

arrangements

Please fill your answer as question 31 of

General Purpose Answer Sheet

ConcepTest 2

You are using a wrench and trying to loosen a

rusty nut. Which of the arrangements shown is

most effective in loosening the nut List in

order of descending efficiency the following

arrangements

Convince your neighbor!

Please fill your answer as question 32 of

General Purpose Answer Sheet

ConcepTest 2

You are using a wrench and trying to loosen a

rusty nut. Which of the arrangements shown is

most effective in loosening the nut List in

order of descending efficiency the following

arrangements

2, 1, 4, 3 or 2, 4, 1, 3

What if two or more different forces act on lever

arm

Net Torque

- The net torque is the sum of all the torques

produced by all the forces - Remember to account for the direction of the

tendency for rotation - Counterclockwise torques are positive
- Clockwise torques are negative

Example 1

N

Determine the net torque

4 m

2 m

Given weights w1 500 N w2 800 N lever

arms d14 m d22 m Find

St

500 N

800 N

1. Draw all applicable forces

2. Consider CCW rotation to be positive

Rotation would be CCW

Where would the 500 N person have to be relative

to fulcrum for zero torque

Example 2

N

y

2 m

d2 m

Given weights w1 500 N w2 800 N lever

arms d14 m St 0 Find d2

500 N

800 N

1. Draw all applicable forces and moment arms

According to our understanding of torque there

would be no rotation and no motion!

What does it say about acceleration and force

Thus, according to 2nd Newtons law SF0 and a0!

Torque and Equilibrium

- First Condition of Equilibrium
- The net external force must be zero
- This is a necessary, but not sufficient,

condition to ensure that an object is in complete

mechanical equilibrium - This is a statement of translational equilibrium
- Second Condition of Equilibrium
- The net external torque must be zero
- This is a statement of rotational equilibrium

Axis of Rotation

- So far we have chosen obvious axis of rotation
- If the object is in equilibrium, it does not

matter where you put the axis of rotation for

calculating the net torque - The location of the axis of rotation is

completely arbitrary - Often the nature of the problem will suggest a

convenient location for the axis - When solving a problem, you must specify an axis

of rotation - Once you have chosen an axis, you must maintain

that choice consistently throughout the problem

Center of Gravity

- The force of gravity acting on an object must be

considered - In finding the torque produced by the force of

gravity, all of the weight of the object can be

considered to be concentrated at one point

Calculating the Center of Gravity

- The object is divided up into a large number of

very small particles of weight (mg) - Each particle will have a set of coordinates

indicating its location (x,y) - The torque produced by each particle about the

axis of rotation is equal to its weight times its

lever arm - We wish to locate the point of application of the

single force , whose magnitude is equal to the

weight of the object, and whose effect on the

rotation is the same as all the individual

particles. - This point is called the center of gravity of the

object

Coordinates of the Center of Gravity

- The coordinates of the center of gravity can be

found from the sum of the torques acting on the

individual particles being set equal to the

torque produced by the weight of the object - The center of gravity of a homogenous, symmetric

body must lie on the axis of symmetry. - Often, the center of gravity of such an object is

the geometric center of the object.

Example

Find center of gravity of the following system

Given masses m1 5.00 kg m2 2.00 kg m3

4.00 kg lever arms d10.500 m d21.00

m Find Center of gravity

Experimentally Determining the Center of Gravity

- The wrench is hung freely from two different

pivots - The intersection of the lines indicates the

center of gravity - A rigid object can be balanced by a single force

equal in magnitude to its weight as long as the

force is acting upward through the objects

center of gravity

Equilibrium, once again

- A zero net torque does not mean the absence of

rotational motion - An object that rotates at uniform angular

velocity can be under the influence of a zero net

torque - This is analogous to the translational situation

where a zero net force does not mean the object

is not in motion

Example of aFree Body Diagram

- Isolate the object to be analyzed
- Draw the free body diagram for that object
- Include all the external forces acting on the

object

Example

Suppose that you placed a 10 m ladder (which

weights 100 N) against the wall at the angle of

30. What are the forces acting on it and when

would it be in equilibrium

Example

Given weights w1 100 N length l10 m angle

a30 St 0 Find f n P

mg

a

1. Draw all applicable forces

2. Choose axis of rotation at bottom corner (t of

f and n are 0!)

Torques Forces

Note f ms n, so

So far net torque was zero. What if it is not

Torque and Angular Acceleration

- When a rigid object is subject to a net torque

(0), it undergoes an angular acceleration - The angular acceleration is directly proportional

to the net torque - The relationship is analogous to F ma
- Newtons Second Law

Torque and Angular Acceleration

torque t

dependent upon object and axis of rotation.

Called moment of inertia I. Units kg m2

The angular acceleration is inversely

proportional to the analogy of the mass in a

rotating system

Example Moment of Inertia of a Uniform Ring

- Image the hoop is divided into a number of small

segments, m1 - These segments are equidistant from the axis

Other Moments of Inertia

Newtons Second Law for a Rotating Object

- The angular acceleration is directly proportional

to the net torque - The angular acceleration is inversely

proportional to the moment of inertia of the

object - There is a major difference between moment of

inertia and mass the moment of inertia depends

on the quantity of matter and its distribution in

the rigid object. - The moment of inertia also depends upon the

location of the axis of rotation

Lets watch the movie!

Example

Consider a flywheel (cylinder pulley) of mass M5

kg and radius R0.2 m and weight of 9.8 N hanging

from rope wrapped around flywheel. What are

forces acting on flywheel and weight Find

acceleration of the weight.

mg

Example

N

T

T

Mg

Given masses M 5 kg weight w 9.8

N radius R0.2 m Find Forces

mg

1. Draw all applicable forces

Forces Torques

Tangential acceleration at the edge of flywheel

(aat)

ConcepTest 3

A force F is applied to a dumbbell for a time

interval Dt, first as in (a) and then as in (b).

In which case does the dumbbell acquire the

greater center-of-mass speed

1. (a) 2. (b) 3. no difference 4. The answer

depends on the rotational inertia of the

dumbbell.

Please fill your answer as question 33 of

General Purpose Answer Sheet

ConcepTest 3

A force F is applied to a dumbbell for a time

interval Dt, first as in (a) and then as in (b).

In which case does the dumbbell acquire the

greater center-of-mass speed

Convince your neighbor!

1. (a) 2. (b) 3. no difference 4. The answer

depends on the rotational inertia of the

dumbbell.

Please fill your answer as question 34 of

General Purpose Answer Sheet

ConcepTest 3

A force F is applied to a dumbbell for a time

interval Dt, first as in (a) and then as in (b).

In which case does the dumbbell acquire the

greater center-of-mass speed

1. (a) 2. (b) 3. no difference 4. The answer

depends on the rotational inertia of the

dumbbell.

Force acts the same time change of momentum is

the same. Thus CM speed is the same as well.

Return to our example

Consider a flywheel (cylinder pulley) of mass M5

kg and radius R0.2 m with weight of 9.8 N

hanging from rope wrapped around flywheel. What

are forces acting on flywheel and weight Find

acceleration of the weight.

mg

If flywheel initially at rest and then begins to

rotate, a torque must be present

Define physical quantity

Angular Momentum

- Similarly to the relationship between force and

momentum in a linear system, we can show the

relationship between torque and angular momentum - Angular momentum is defined as L I
- If the net torque is zero, the angular momentum

remains constant - Conservation of Linear Momentum states The

angular momentum of a system is conserved when

the net external torque acting on the systems is

zero. - That is, when

(compare to )

Return to our example once again

Consider a flywheel (cylinder pulley) of mass M5

kg and radius R0.2 m with weight of 9.8 N

hanging from rope wrapped around flywheel. What

are forces acting on flywheel and weight Find

acceleration of the weight.

mg

Each small part of the flywheel is moving with

some velocity. Therefore, each part and the

flywheel as a whole have kinetic energy!

Thus, total KE of the system

Total Energy of Rotating System

- An object rotating about some axis with an

angular speed, , has rotational kinetic energy

½I2 - Energy concepts can be useful for simplifying the

analysis of rotational motion - Conservation of Mechanical Energy
- Remember, this is for conservative forces, no

dissipative forces such as friction can be present

ConcepTest 4

A force F is applied to a dumbbell for a time

interval Dt, first as in (a) and then as in (b).

In which case does the dumbbell acquire the

greater energy

1. (a) 2. (b) 3. no difference 4. The answer

depends on the rotational inertia of the

dumbbell.

Please fill your answer as question 35 of

General Purpose Answer Sheet

ConcepTest 4

A force F is applied to a dumbbell for a time

interval Dt, first as in (a) and then as in (b).

In which case does the dumbbell acquire the

greater energy

Convince your neighbor!

1. (a) 2. (b) 3. no difference 4. The answer

depends on the rotational inertia of the

dumbbell.

Please fill your answer as question 36 of

General Purpose Answer Sheet

ConcepTest 4

A force F is applied to a dumbbell for a time

interval Dt, first as in (a) and then as in (b).

In which case does the dumbbell acquire the

greater energy

1. (a) 2. (b) 3. no difference 4. The answer

depends on the rotational inertia of the

dumbbell.

Since CM speeds are the same, translational

kinetic energies are the same. But (b) also

rotates, so it also has rotational energy.

Note on problem solving

- The same basic techniques that were used in

linear motion can be applied to rotational

motion. - Analogies F becomes , m becomes I and a

becomes , v becomes and x becomes - Techniques for conservation of energy are the

same as for linear systems, as long as you

include the rotational kinetic energy - Problems involving angular momentum are

essentially the same technique as those with

linear momentum - The moment of inertia may change, leading to a

change in angular momentum

Review before Exam 2

- Useful tips
- Do and understand all the homework problems.
- Review and understand all the problems done in

class. - Review and understand all the problems done in

the textbook (chapters 5-8). - Come to office hours if you have questions!!!

Exam Review

- Chapter 5 (Work and energy)
- Work, kinetic and potential energy. Reference

levels. Elastic energy. - Conservation of energy. Power.
- Chapter 6 (Momentum and collisions)
- Impulse-momentum theorem. Conservation of

momentum. - Chapter 7 (Circular motion and gravity)
- Angular velocity and acceleration. Centripetal

acceleration. - Newtons law of universal gravitation.
- Chapter 8 (Rotational dynamics)

Review problem

The launching mechanism of a toy gun consists of

a spring of unknown spring constant, as shown in

Figure 1. If the spring is compressed a distance

of 0.120 m and the gun fired vertically as shown,

the gun can launch a 20.0-g projectile from rest

to a maximum height of 20.0 m above the starting

point of the projectile. Neglecting all resistive

forces, determine (a) the spring constant and (b)

the speed of the projectile as it moves through

the equilibrium position of the spring (where x

0), as shown in Figure 1.

(No Transcript)

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