Chapter 8 Torque and Angular Momentum MFMc Graw

Chapter 8 Torque and Angular Momentum MFMc. Graw Chap 08 -Torque-Revised 3/6/2010

Torque and Angular Momentum • Rotational Kinetic Energy • Rotational Inertia • Torque • Work Done by a Torque • Equilibrium (revisited) • Rotational Form of Newton’s 2 nd Law • Rolling Objects • Angular Momentum MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 2

Rotational KE and Inertia For a rotating solid body: For a rotating body vi = ri where ri is the distance from the rotation axis to the mass mi. MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 3

Moment of Inertia The quantity is called rotational inertia or moment of inertia. Use the above expression when the number of masses that make up a body is small. Use the moments of inertia in the table in the textbook for extended bodies. MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 4

Moments of Inertia MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 5

MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 6

Moment of Inertia Example: The masses are m 1 and m 2 and they are separated by a distance r. Assume the rod connecting the masses is massless. Q: (a) Find the moment of inertia of the system below. m 1 MFMc. Graw r 1 r 2 m 2 r 1 and r 2 are the distances between mass 1 and the rotation axis and mass 2 and the rotation axis (the dashed, vertical line) respectively. Chap 08 -Torque-Revised 3/6/2010 7

Moment of Inertia Take m 1 = 2. 00 kg, m 2 = 1. 00 kg, r 1= 0. 33 m , and r 2 = 0. 67 m. (b) What is the moment of inertia if the axis is moved so that is passes through m 1? MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 8

Moment of Inertia What is the rotational inertia of a solid iron disk of mass 49. 0 kg with a thickness of 5. 00 cm and a radius of 20. 0 cm, about an axis through its center and perpendicular to it? From the table: MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 9

Torque A torque is caused by the application of a force, on an object, at a point other than its center of mass or its pivot point. hinge Q: Where on a door do you normally push to open it? P u s h A: Away from the hinge. A rotating (spinning) body will continue to rotate unless it is acted upon by a torque. MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 10

Torque method 1: F Top view of door Hinge end r = the distance from the rotation axis (hinge) to the point where the force F is applied. F is the component of the force F that is perpendicular to the door (here it is Fsin ). MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 11

Torque The units of torque are Newton-meters (Nm) (not joules!). By convention: • When the applied force causes the object to rotate counterclockwise (CCW) then is positive. • When the applied force causes the object to rotate clockwise (CW) then is negative. MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 12

Torque method 2: r is called the lever arm and F is the magnitude of the applied force. Lever arm is the perpendicular distance to the line of action of the force. MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 13

Torque F Top view of door Hinge end r Line of action of the force Lever arm The torque is: Same as before MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 14

Torque Problem The pull cord of a lawnmower engine is wound around a drum of radius 6. 00 cm, while the cord is pulled with a force of 75. 0 N to start the engine. What magnitude torque does the cord apply to the drum? F=75 N R=6. 00 cm MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 15

Torque Problem Calculate the torque due to the three forces shown about the left end of the bar (the red X). The length of the bar is 4 m and F 2 acts in the middle of the bar. F 2=30 N 30 F 3=20 N 45 10 X F 1=25 N MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 16

Torque Problem Lever arm for F 2 45 F 2=30 N 30 F 3=20 N 10 X F 1=25 N Lever arm for F 3 The lever arms are: MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 17

Torque Problem The torques are: The net torque is +65. 8 Nm and is the sum of the above results. MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 18

Work done by the Torque The work done by a torque is where is the angle (in radians) that the object turns through. Following the analogy between linear and rotational motion: Linear Work is Force x displacement. In the rotational picture force becomes torque and displacement becomes the angle MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 19

Work done by the Torque A flywheel of mass 182 kg has a radius of 0. 62 m (assume the flywheel is a hoop). (a) What is the torque required to bring the flywheel from rest to a speed of 120 rpm in an interval of 30 sec? MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 20

Work done by the Torque (b) How much work is done in this 30 sec period? MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 21

Equilibrium The conditions for equilibrium are: Linear motion Rotational motion For motion in a plane we now have three equations to satisfy. MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 22

Using Torque A sign is supported by a uniform horizontal boom of length 3. 00 m and weight 80. 0 N. A cable, inclined at a 35 angle with the boom, is attached at a distance of 2. 38 m from the hinge at the wall. The weight of the sign is 120. 0 N. What is the tension in the cable and what are the horizontal and vertical forces exerted on the boom by the hinge? MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 23

Using Torque This is important! You need two components for F, not just the expected perpendicualr normal force. FBD for the bar: y Fy T X Fx x wbar Fsb Apply the conditions for equilibrium to the bar: MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 24

Using Torque Equation (3) can be solved for T: Equation (1) can be solved for Fx: Equation (2) can be solved for Fy: MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 25

Equilibrium in the Human Body Find the force exerted by the biceps muscle in holding a one liter milk carton with the forearm parallel to the floor. Assume that the hand is 35. 0 cm from the elbow and that the upper arm is 30. 0 cm long. The elbow is bent at a right angle and one tendon of the biceps is attached at a position 5. 00 cm from the elbow and the other is attached 30. 0 cm from the elbow. The weight of the forearm and empty hand is 18. 0 N and the center of gravity is at a distance of 16. 5 cm from the elbow. MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 26

MCAT type problem y Fb “hinge” (elbow joint) x w MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 Fca 27

Newton’s 2 nd Law in Rotational Form Compare to MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 28

Rolling Object A bicycle wheel (a hoop) of radius 0. 3 m and mass 2 kg is rotating at 4. 00 rev/sec. After 50 sec the wheel comes to a stop because of friction. What is the magnitude of the average torque due to frictional forces? MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 29

Rolling Objects An object that is rolling combines translational motion (its center of mass moves) and rotational motion (points in the body rotate around the center of mass). For a rolling object: If the object rolls without slipping then vcm = R. Note the similarity in the form of the two kinetic energies. MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 30

Rolling Example Two objects (a solid disk and a solid sphere) are rolling down a ramp. Both objects start from rest and from the same height. Which object reaches the bottom of the ramp first? h This we know - The object with the largest linear velocity (v) at the bottom of the ramp will win the race. MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 31

Rolling Example Apply conservation of mechanical energy: Solving for v: MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 32

Rolling Example continued: Note that the mass and radius are the same. The moments of inertia are: For the disk: Since Vsphere> Vdisk the sphere wins the race. For the sphere: Compare these to a box sliding down the ramp. MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 33

The Disk or the Ring? MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 34

How do objects in the previous example roll? Both the normal force and the weight act through the center of mass so FBD: y N = 0. This means that the object cannot rotate when only these two forces are applied. w x The round object won’t rotate, but most students have difficulty imagining a sphere that doesn’t rotate when moving down hill. MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 35

Add Friction y FBD: N Fs w Also need acm = R and x The above system of equations can be solved for v at the bottom of the ramp. The result is the same as when using energy methods. (See text example 8. 13. ) It is the addition of static friction that makes an object roll. MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 36

Angular Momentum Units of p are kg m/s When no net external forces act, the momentum of a system remains constant (pi = pf) MFMc. Graw Units of L are kg m 2/s When no net external torques act, the angular momentum of a system remains constant (Li = Lf). Chap 08 -Torque-Revised 3/6/2010 37

Angular Momentum Units of p are kg m/s When no net external forces act, the momentum of a system remains constant (pi = pf) MFMc. Graw Units of L are kg m 2/s When no net external torques act, the angular momentum of a system remains constant (Li = Lf). Chap 08 -Torque-Revised 3/6/2010 38

Angular Momentum Example A turntable of mass 5. 00 kg has a radius of 0. 100 m and spins with a frequency of 0. 500 rev/sec. Assume a uniform disk. What is the angular momentum? MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 39

Angular Momentum Example A skater is initially spinning at a rate of 10. 0 rad/sec with I=2. 50 kg m 2 when her arms are extended. What is her angular velocity after she pulls her arms in and reduces I to 1. 60 kg m 2? The skater is on ice, so we can ignore external torques. MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 40

The Vector Nature of Angular Momentum Angular momentum is a vector. Its direction is defined with a right-hand rule. MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 41

The Right-Hand Rule Curl the fingers of your right hand so that they curl in the direction a point on the object moves, and your thumb will point in the direction of the angular momentum. Angular Momentum is also an example of a vector cross product MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 42

The Vector Cross Product The magnitude of C C = ABsin(Φ) The direction of C is perpendicular to the plane of A and B. Physically it means the product of A and the portion of B that is perpendicular to A. MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 43

The Cross Product by Components Since A and B are in the x-y plane A x B is along the z-axis. MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 44

Memorizing the Cross Product MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 45

The Gyroscope Demo MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 46

Angular Momentum Demo Consider a person holding a spinning wheel. When viewed from the front, the wheel spins CCW. Holding the wheel horizontal, they step on to a platform that is free to rotate about a vertical axis. MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 47

Angular Momentum Demo Initially, nothing happens. They then move the wheel so that it is over their head. As a result, the platform turns CW (when viewed from above). This is a result of conserving angular momentum. MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 48

Angular Momentum Demo Initially there is no angular momentum about the vertical axis. When the wheel is moved so that it has angular momentum about this axis, the platform must spin in the opposite direction so that the net angular momentum stays zero. Is angular momentum conserved about the direction of the wheel’s initial, horizontal axis? MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 49

It is not. The floor exerts a torque on the system (platform + person), thus angular momentum is not conserved here. MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 50

Summary • Rotational Kinetic Energy • Moment of Inertia • Torque (two methods) • Conditions for Equilibrium • Newton’s 2 nd Law in Rotational Form • Angular Momentum • Conservation of Angular Momentum MFMc. Graw Chap 08 -Torque-Revised 3/6/2010 51

X Z MFMc. Graw Y Chap 08 -Torque-Revised 3/6/2010 52