Mastering Rotational Dynamics: Solving Problems and Understanding Angular Motion in Physics 253

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2. Chapter 10: Rotational Motional About a Fixed Axis • 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) Physics 253

3. Review of Quiz 3 • Points • Total: 50 • Average: 35 • High: 50 • Low: 6 • Score Distribution • 00-10: 1 • 11-20: 3 • 21-30: 22 • 31-40: 42 • 41-50: 24 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) 2N b) -98.0N c) 4.9 J d) 49 J • Answer: The force and displacement are in the same direction so W=(F)(d)=(mg)(d)= (10.0kg)(9.8m/s2)(0.50m)=49J 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: Physics 253

6. Problem 3 • Problem 3 (5 points): What is the center of mass for m1 = 5.0 kg, x1 = 2.0 m, m2 = 12 kg, x2 = 6.0 m? a) 82m b) 4.8 m c) 8.5m d)0.021m • Answer: Physics 253

7. Problem 4 • Problem 4 (5 points): A spring compressed by 0.10 m stores 30J of energy. What is the spring constant? a) 6000N/m b) 600 N/m c) 300N/m d) 0.03N/m • Answer: 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 100.0kg and a velocity of +2.5m/s. If the second astronaut has a velocity of -4.0m/s, what is her mass? • Answer: 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: Physics 253

10. Problem 7 • Problem 7 (15 points): ): A 5x104 kg spaceship is traveling at a speed of 1.1x104 m/s. The engine exerts a force of 4x105N parallel to the displacement and fires until the displacement is 2.5x106 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: Physics 253

11. Review Torque • Torque = (Lever Arm) x (Magnitude of the force) 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 2nd Law. 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 mR2 and represents the rotational inertia of the particle and is often called the moment of inertia. 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=SmiRi2 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 Physics 253

16. Two small weights of mass 5.0 and 7.0kg are mounted 4.0m 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 5.0kg mass Calculate the force on the 7.0kg mass needed to achieve an acceleration of 1 rad/sec2 Example 1: Moments of Inertia Calculations 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. Physics 253

19. Example 2: A Heavy Pulley A 15.0 N force is applied to a cord around a 4.00kg pulley at a radius of 33.0cm. The pulley accelerates uniformly from rest to 30.0 rad/s in 3.00s. If there is a frictional torque at the axel of 1.10mN what is the pulley’s moment of inertia? 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 15.0N (m=1.53kg) 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 3.00s. 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 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. 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 2nd Laws • Rotational motion: St=Ia • Translational motion: SF=ma • Solve the set of equations for any missing info. • Smell test! Physics 253

29. To Summarize • Ok we’ve defined torque • This led to the analog of Newton’s 2nd 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. Physics 253