Force is applied at the handle The axis of rotation is at the nut or bolt The distance away is the length of the handle. Torque direction is either clockwise (cw) or counterclockwise (ccw) T=F*L Torque = force * lever arm.

LEVER ARM Always the shortest distance from the rotation axis (axle) to the line of action (applied force).” Always the shortest distance from the rotation axis (axle) to the line of action (applied force).”

WHAT HAPPENS TO THE FORCE IS THE HANDLE LENGTH IS DOUBLED?

TORQUE AT EQUILIBRIUM Just like linear forces, torque is at equilibrium if the clockwise torque is equal to the counter clockwise torque. Just like linear forces, torque is at equilibrium if the clockwise torque is equal to the counter clockwise torque. The net force is 0 The net force is 0 This means the object will stay at rest (if it was at rest to begin with) or This means the object will stay at rest (if it was at rest to begin with) or It will continue its original motion (if it was in motion before the torque was applied) It will continue its original motion (if it was in motion before the torque was applied) If they are unequal, then there is a net force in either the clockwise or counterclockwise direction. If they are unequal, then there is a net force in either the clockwise or counterclockwise direction. This means there is a change in the movement. The object will either speed up or slow down its rotation This means there is a change in the movement. The object will either speed up or slow down its rotation

What does this setup remind you of?

Units of Chapter 10 Angular Position, Velocity, and Acceleration Rotational Kinematics Connections Between Linear and Rotational Quantities Rolling Motion Rotational Kinetic Energy and the Moment of Inertia Conservation of Energy

10-1 Angular Position, Velocity, and Acceleration

Degrees and revolutions:

10-1 Angular Position, Velocity, and Acceleration Arc length s, measured in radians: This means

10-1 Angular Position, Velocity, and Acceleration

Calculus: it just means how fast it is turning at the instant that we are talking about. (Just like the speedometer on your car says how fast you are going at the instant)

10-1 Angular Position, Velocity, and Acceleration How long until you make a full circle

10-1 Angular Position, Velocity, and Acceleration Calculus: it just means how fast it is changing speed at the instant that we are talking about. (Just like the speedometer on your car says how fast you are going at the instant)

10-2 Rotational Kinematics Analogies between linear and rotational kinematics:

An object at rest begins to rotate with a constant angular acceleration. If this object has angular velocity  at time t, what was its angular velocity at the time 1/2 t? 1) 1/2  2) 1/4  3) 3/4  4) 2  5) 4  Angular Velocity

An object at rest begins to rotate with a constant angular acceleration. If this object has angular velocity  at time t, what was its angular velocity at the time 1/2 t? 1) 1/2  2) 1/4  3) 3/4  4) 2  5) 4  half the time half the speed The angular velocity is  =  t (starting from rest), and there is a linear dependence on time. Therefore, in half the time, the object has accelerated up to only half the speed. Angular Velocity

EXAMPLE The angular speed of a propeller on a boat increases with constant acceleration from 12 rad/s to 26 rad/s in 2.5 revolutions. The angular speed of a propeller on a boat increases with constant acceleration from 12 rad/s to 26 rad/s in 2.5 revolutions. What is the acceleration of the propeller? What is the acceleration of the propeller? How long did the change in angular speed take? How long did the change in angular speed take?

10-3 Connections Between Linear and Rotational Quantities

TANGENTIAL SPEED

10-3 Connections Between Linear and Rotational Quantities

This merry-go- round has both tangential and centripetal acceleration.

BONNIE AND KLYDE Bonnie Klyde 1) Klyde 2) Bonnie 3) both the same 4) linear velocity is zero for both of them Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one revolution every two seconds. Who has the larger linear (tangential) velocity?

linear speeds v v = R  Bonnie is located further out Their linear speeds v will be different since v = R  and Bonnie is located further out (larger radius R) than Klyde. Bonnie Klyde 1) Klyde 2) Bonnie 3) both the same 4) linear velocity is zero for both of them BONNIE AND KLYDE Follow-up: Who has the larger centripetal acceleration? Bonnie sits on the outer rim of a merry-go- round, and Klyde sits midway between the center and the rim. The merry-go-round makes one revolution every two seconds. Who has the larger linear (tangential) velocity?

Units of Chapter 7 Work Done by a Constant Force Kinetic Energy and the Work-Energy Theorem Work Done by a Variable Force Power

7-1 Work Done by a Constant Force The definition of work, when the force is in the direction of the displacement: (7-1) SI unit: newton-meter (N·m) = joule, J

7-1 Work Done by a Constant Force If the force is at an angle to the displacement: (7-3)

The Sign of Work The work done may be positive, zero, or negative, depending on the angle between the force and the displacement:

TOTAL WORK Sum of the works from each force Sum of the works from each force Or: Work done by Net Force Or: Work done by Net Force

Is it possible to do work on an object that remains at rest? 1) yes 2) no TO WORK OR NOT TO WORK

Is it possible to do work on an object that remains at rest? 1) yes 2) no force acts over a distance no displacementno work done Work requires that a force acts over a distance. If an object does not move at all, there is no displacement, and therefore no work done. TO WORK OR NOT TO WORK

NORMAL FORCE AND WORK 1) Normal force does no work at all 2) Normal force does negative work 3) Normal force does positive work A box is being pulled across a rough floor at a constant speed. What can you say about the work done by the normal force?

f N mg displacement Pull zero W = F d cos  = 90 o W = 0 The normal force is perpendicular to the displacement, so the work is zero. Or using the definition of work (W = F d cos  ), since  = 90 o, then W = 0. NORMAL FORCE AND WORK 1) Normal force does no work at all 2) Normal force does negative work 3) Normal force does positive work A box is being pulled across a rough floor at a constant speed. What can you say about the work done by friction?

FRICTION AND WORK 1) friction does no work at all 2) friction does negative work 3) friction does positive work A box is being pulled across a rough floor at a constant speed. What can you say about the work done by friction?

f N mg displacement Pull opposite negative W = F d cos  = 180 o W < 0 Friction acts in the opposite direction to the displacement, so the work is negative. Or using the definition of work (W = F d cos  ), since  = 180 o, then W < 0. FRICTION AND WORK 1) friction does no work at all 2) friction does negative work 3) friction does positive work A box is being pulled across a rough floor at a constant speed. What can you say about the work done by friction?

FORCE AND WORK 1) one force 2) two forces 3) three forces 4) four forces 5) no forces are doing work A box is being pulled up a rough incline by a rope connected to a pulley. How many forces are doing work on the box?

FORCE AND WORK N f T mg displacement Any force not perpendicular to the motion will do work: no work N does no work positive T does positive work f does negative work mg does negative work 1) one force 2) two forces 3) three forces 4) four forces 5) no forces are doing work A box is being pulled up a rough incline by a rope connected to a pulley. How many forces are doing work on the box?

7-2 Kinetic Energy and the Work-Energy Theorem When positive work is done on an object, its speed increases; when negative work is done, its speed decreases.

Kinetic Energy After algebraic manipulations of the equations of motion, we find: Therefore, we define the kinetic energy: (7-6)

The Work-Energy Theorem Work-Energy Theorem: The total work done on an object is equal to its change in kinetic energy. (7-7)

A child on a skateboard is moving at a speed of 2 m/s. After a force acts on the child, her speed is 3 m/s. What can you say about the work done by the external force on the child? 1) positive work was done 2) negative work was done 3) zero work was done CONCEPTEST 7.7 WORK AND KE

A child on a skateboard is moving at a speed of 2 m/s. After a force acts on the child, her speed is 3 m/s. What can you say about the work done by the external force on the child? 1) positive work was done 2) negative work was done 3) zero work was done The kinetic energy of the child increased because her speed increasedincrease in KEpositive work being done KE f > KE i work W must be positive The kinetic energy of the child increased because her speed increased. This increase in KE was the result of positive work being done. Or, from the definition of work, since W =  KE = KE f – KE i and we know that KE f > KE i in this case, then the work W must be positive. CONCEPTEST 7.7 WORK AND KE Follow-up: What does it mean for negative work to be done on the child?

CONCEPTEST 7.8B SPEEDING UP I 1) 0  30 mph 2) 30  60 mph 3) both the same A car starts from rest and accelerates to 30 mph. Later, it gets on a highway and accelerates to 60 mph. Which takes more energy, the 0  30 mph, or the 30  60 mph? A car starts from rest and accelerates to 30 mph. Later, it gets on a highway and accelerates to 60 mph. Which takes more energy, the 0  30 mph, or the 30  60 mph?

1/2 mv 2 velocity squared The change in KE (1/2 mv 2 ) involves the velocity squared. 1/2 m ( ) = 1/2 m (900) So in the first case, we have: 1/2 m ( ) = 1/2 m (900) 1/2 m ( ) = 1/2 m (2700) In the second case, we have: 1/2 m ( ) = 1/2 m (2700) bigger energy changesecond case Thus, the bigger energy change occurs in the second case. CONCEPTEST 7.8B SPEEDING UP I 1) 0  30 mph 2) 30  60 mph 3) both the same A car starts from rest and accelerates to 30 mph. Later, it gets on a highway and accelerates to 60 mph. Which takes more energy, the 0  30 mph, or the 30  60 mph? A car starts from rest and accelerates to 30 mph. Later, it gets on a highway and accelerates to 60 mph. Which takes more energy, the 0  30 mph, or the 30  60 mph?

EXAMPLE A 1300 kg car coasts on a horizontal road with a velocity of 18 m/s, E. After crossing an unpaved, sandy stretch of road 30.0 m long, it’s velocity decreases to 15 m/s, E. A 1300 kg car coasts on a horizontal road with a velocity of 18 m/s, E. After crossing an unpaved, sandy stretch of road 30.0 m long, it’s velocity decreases to 15 m/s, E. Was the net work done on the car positive, negative, or zero? Was the net work done on the car positive, negative, or zero? Find the net work done on the car. Find the net work done on the car. What is the magnitude and direction of the average net force on the car in the sandy section? What is the magnitude and direction of the average net force on the car in the sandy section?

7-3 Work Done by a Variable Force If the force is constant, we can interpret the work done graphically:

Multiple Rectangles If the force takes on several successive constant values:

Varying Force We can then approximate a continuously varying force by a succession of constant values.

Work done by a spring The force needed to stretch a spring an amount x is F = kx. Therefore, the work done in stretching the spring is (7-8)

EXAMPLE A 1.2 kg block is held against a spring of force constant 1.0x10 4 N/m, compressing it a distance of 0.15 m. How fast is the block moving after it has been released and the spring pushes it away (in other words, just after the spring stops pushing on it)? A 1.2 kg block is held against a spring of force constant 1.0x10 4 N/m, compressing it a distance of 0.15 m. How fast is the block moving after it has been released and the spring pushes it away (in other words, just after the spring stops pushing on it)?

7-4 Power Power is a measure of the rate at which work is done: (7-10) SI unit: J/s = watt, W 1 horsepower = 1 hp = 746 W

7-4 Power

PAYING YOUR ELECTRIC BILL

Engine #1 produces twice the power of engine #2. Can we conclude that engine #1 does twice as much work as engine #2? 1) yes 2) no CONCEPTEST 7.11C POWER

Engine #1 produces twice the power of engine #2. Can we conclude that engine #1 does twice as much work as engine #2? 1) yes 2) no No!! We cannot conclude anything about how much work each engine does.work will depend upon how much time is used No!! We cannot conclude anything about how much work each engine does. Given the power output, the work will depend upon how much time is used. For example, engine #1 may do the same amount of work as engine #2, but in half the time. CONCEPTEST 7.11C POWER

CONCEPTEST 7.12B ENERGY CONSUMPTION Which contributes more to the cost of your electric bill each month, a 1500-Watt hair dryer or a 600-Watt microwave oven? 1) hair dryer 2) microwave oven 3) both contribute equally 4) depends upon what you cook in the oven 5) depends upon how long each one is on 1500 W 600 W

energy you have to know how long it was running We already saw that what you actually pay for is energy. To find the energy consumption of an appliance, you must know more than just the power rating — you have to know how long it was running. CONCEPTEST 7.12B ENERGY CONSUMPTION Which contributes more to the cost of your electric bill each month, a 1500-Watt hair dryer or a 600-Watt microwave oven? 1) hair dryer 2) microwave oven 3) both contribute equally 4) depends upon what you cook in the oven 5) depends upon how long each one is on 1500 W 600 W

7-4 Power If an object is moving at a constant speed in the face of friction, gravity, air resistance, and so forth, the power exerted by the driving force can be written: (7-13)

Summary of Chapter 7 If the force is constant and parallel to the displacement, work is force times distance If the force is not parallel to the displacement, The total work is the work done by the net force:

Summary of Chapter 7 SI unit of work: the joule, J Total work is equal to the change in kinetic energy: where

Summary of Chapter 7 Work done by a spring force: Power is the rate at which work is done: SI unit of power: the watt, W