1 3102021 Physics 253 2 Chapter 10 Rotational

2 Chapter 10: Rotational Motional About a Fixed Axis ü Review of Angular Quantities

3 Review of Quiz 3 • Points – – Total: 50 Average: 35 High:

4 Problem 1 • Problem 1 (5 points): How much work must be supplied

5 Problem 2 • Problem 2 (5 points): The earth’s gravitational potential energy is

6 Problem 3 • Problem 3 (5 points): What is the center of mass

7 Problem 4 • Problem 4 (5 points): A spring compressed by 0. 10

8 Problem 5 • Problem 5 (5 points): Two astronauts in outer space and

9 Problem 6 • Problem 6 (10 points): A motorcyclist is trying to leap

10 Problem 7 • • Problem 7 (15 points): ): A 5 x 104

11 Review Torque • Torque = (Lever Arm) x (Magnitude of the force) 3/10/2021

12 Rotational Dynamics: Torque and Rotational Inertia • Just as force is proportional to

13 • Consider a particle of mass m rotating on a circle of radius

15 Comments on Moment of Inertia • Serves the same role for rotational motion

16 Example 1: Moments of Inertia Calculations • Two small weights of mass 5.

18 Characteristics of the Moment of Inertia • Different for different axes. • Masses

19 Example 2: A Heavy Pulley A 15. 0 N force is applied to

23 Example 3: A Pulley and Bucket • Lets take the heavy pulley problem

24 • Our strategy will be to – analyze the rotational motion of the

26 Example 4: A Rotating Rod • A uniform rod of mass M and

28 Solving Problems with Rotational Motion • Draw the free-body diagram! – Show all

29 To Summarize • Ok we’ve defined torque • This led to the analog

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1 3/10/2021 Physics 253

2 Chapter 10: Rotational Motional About a Fixed Axis ü Review of Angular Quantities & Motion and Torque (10 -1, 10 -2, 10 -3, 10 -5) • Solving Problems in Rotational Dynamics (10 -6, 10 -7) • Determining Moments of Inertia, Conservation of Angular Momentum, and Rotational Kinetic Energy (10 -8, 10 -9, 10 -10) • Rotational + Translational Motion (10 -11, 10 -12) 3/10/2021 Physics 253

3 Review of Quiz 3 • Points – – Total: 50 Average: 35 High: 50 Low: 6 – – – 00 -10: 1 11 -20: 3 21 -30: 22 31 -40: 42 41 -50: 24 • Score Distribution 3/10/2021 Average: 36 Physics 253

4 Problem 1 • Problem 1 (5 points): How much work must be supplied to vertically lift a 10. 0 -kg box 0. 50 meters? a) 2 N b) -98. 0 N c) 4. 9 J d) 49 J • Answer: The force and displacement are in the same direction so W=(F)(d)=(mg)(d)= (10. 0 kg)(9. 8 m/s 2)(0. 50 m)=49 J 3/10/2021 Physics 253

5 Problem 2 • Problem 2 (5 points): The earth’s gravitational potential energy is defined to be zero at: a) the earth’s surface b) infinite distance c) near the moon d) the earth’s center • Answer: 3/10/2021 Physics 253

6 Problem 3 • Problem 3 (5 points): What is the center of mass for m 1 = 5. 0 kg, x 1 = 2. 0 m, m 2 = 12 kg, x 2 = 6. 0 m? a) 82 m b) 4. 8 m c) 8. 5 m d)0. 021 m • Answer: 3/10/2021 Physics 253

7 Problem 4 • Problem 4 (5 points): A spring compressed by 0. 10 m stores 30 J of energy. What is the spring constant? a) 6000 N/m b) 600 N/m c) 300 N/m d) 0. 03 N/m • Answer: 3/10/2021 Physics 253

8 Problem 5 • Problem 5 (5 points): Two astronauts in outer space and initially at rest “push off” and move away from one another. The first astronaut has a mass of 100. 0 kg and a velocity of +2. 5 m/s. If the second astronaut has a velocity of -4. 0 m/s, what is her mass? • Answer: 3/10/2021 Physics 253

9 Problem 6 • Problem 6 (10 points): A motorcyclist is trying to leap across a canyon as shown in the figure. When he leaves the cliff the cycle has a speed of 38. 0 m/s. Using conservation of energy find the speed with which the cycle strikes the ground on the other side. • Answer: 3/10/2021 Physics 253

10 Problem 7 • • Problem 7 (15 points): ): A 5 x 104 kg spaceship is traveling at a speed of 1. 1 x 104 m/s. The engine exerts a force of 4 x 105 N parallel to the displacement and fires until the displacement is 2. 5 x 106 m. (No forces act on the vessel except that generated by its engine. ) Determine (a) the initial kinetic energy, b) the work done by the engine and c) using the Work-Energy Theorem the final velocity of the spaceship. Answer: 3/10/2021 Physics 253

11 Review Torque • Torque = (Lever Arm) x (Magnitude of the force) 3/10/2021 Physics 253

12 Rotational Dynamics: Torque and Rotational Inertia • Just as force is proportional to acceleration, torque seems to be proportional to angular acceleration: • Note that the constant of proportionality for linear motion is mass. Just on dimensional grounds the analogous constant must be different for rotational motion. • Turns out we can get there starting with the 2 nd Law. 3/10/2021 Physics 253

13 • Consider a particle of mass m rotating on a circle of radius r at the end of a massless string or rod and subject to a force F. • The constant of proportionality is m. R 2 and represents the rotational inertia of the particle and is often called the moment of inertia. 3/10/2021 Physics 253

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15 Comments on Moment of Inertia • Serves the same role for rotational motion as mass does for linear motion • But since I=Smi. Ri 2 is a sum over many objects it depends on mass distribution – If of equal mass, a larger cylinder will have a greater moment of inertia than a smaller one. Something intuitively true. – When mass is far from the axis, it also hard to rotate something, again something familiar. – For rotational motion the mass of a body cannot be considered as concentrated at the center of mass. • Still it can be extended to the center of mass 3/10/2021 Physics 253

16 Example 1: Moments of Inertia Calculations • Two small weights of mass 5. 0 and 7. 0 kg are mounted 4. 0 m apart on a massless rod. • Calculate the moment of inertia I – About an axis halfway between the weights – About an axis 0. 50 meter to left of the 5. 0 kg mass • Calculate the force on the 7. 0 kg mass needed to achieve an acceleration of 1 rad/sec 2 3/10/2021 Physics 253

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18 Characteristics of the Moment of Inertia • Different for different axes. • Masses close to the axis contribute little, but masses distant contribute much. • Calculations can be difficult because the mass distributions are not uniform. • They can be worked out taking the sum to the limit and using calculus: • Experimentally done by measuring a for a known t. 3/10/2021 Physics 253

19 Example 2: A Heavy Pulley A 15. 0 N force is applied to a cord around a 4. 00 kg pulley at a radius of 33. 0 cm. The pulley accelerates uniformly from rest to 30. 0 rad/s in 3. 00 s. If there is a frictional torque at the axel of 1. 10 m. N what is the pulley’s moment of inertia? 3/10/2021 Physics 253

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23 Example 3: A Pulley and Bucket • Lets take the heavy pulley problem a bit further and hang a bucket of weight 15. 0 N (m=1. 53 kg) from it. • Calculate – The angular acceleration of the pulley, a, and the linear acceleration of the bucket, a. – If the pulley and bucket start at rest calculate the angular velocity of the pulley, w, and the linear velocity of the bucket, v, at 3. 00 s. 3/10/2021 Physics 253

24 • Our strategy will be to – analyze the rotational motion of the pulley – analyze the linear motion of the bucket – Connect the two and solve for accelerations – Use accelerations to find the final velocity 3/10/2021 Physics 253

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26 Example 4: A Rotating Rod • A uniform rod of mass M and length L can pivot freely up or down. The rod is held horizontally and then released. • Assuming the force of gravity acts at the center of mass, at the moment of release determine – The angular acceleration of the rod – The linear acceleration at the tip of the rod. 3/10/2021 Physics 253

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28 Solving Problems with Rotational Motion • Draw the free-body diagram! – Show all forces and WHERE they act. – Put gravity at the CG or CM. • Identify the axis of rotation and calculate torques about it. – CCW is positive – CW is negative. • Apply the 2 nd Laws – Rotational motion: St=Ia – Translational motion: SF=ma • Solve the set of equations for any missing info. • Smell test! 3/10/2021 Physics 253

29 To Summarize • Ok we’ve defined torque • This led to the analog of Newton’s 2 nd Law for angular motion: • Where the moment of inertia serves the role of mass • Next we’ll learn how to calculate I and explore angular momentums and rotational energy. 3/10/2021 Physics 253