ML 3 1. The moment of inertia of a long rod spun around an axis through one end perpendicular to its length is Why is this moment of inertia greater than it would be if you spun a point mass M at the location of the center of mass of the rod (at ML )? (That would be ) 2. Why is the moment of inertia of a hoop that has a mass M and a radius R greater than the moment of inertia of a disk that has the same mass and radius? Why is the moment of inertia of a spherical shell that has a mass M and a radius R greater than that of a solid sphere that has the same mass and radius? 3. Give an example in which a small force exerts a large torque. Give another example in which a large force exerts a small torque. 4. While reducing the mass of a racing bike, the greatest benefit is realized from reducing the mass of the tires and wheel rims. Why does this allow a racer to achieve greater accelerations than would an identical reduction in the mass of the bicycle's frame? 5. Describe the energy transformations involved when a yo-yo is thrown downward and then climbs back up its string to be caught in the user's hand. 9 6. Calculate the moment of inertia of a skater given the following information. (a) The 60.0-kg skater is approximated as a cylinder that has a 0.110-m radius. (b) The skater with arms extended is approximately a cylinder that is 52.5 kg, has a 0.110-m radius, and has two 0.900-m-long arms which are 3.75 kg each and extend straight out from the cylinder like rods rotated about their ends. (OpenStax 10.11) 0.363 kg. m, 2.34 kg m . 7. The triceps muscle in the back of the upper arm extends the forearm. This muscle in a professional boxer exerts a force of 2.00 103 N with an effective perpendicular lever arm of 3.00 cm, producing an angular acceleration of the forearm of 120 rad/s. What is the moment of inertia of the boxer's forearm? (OpenStax 10.12) 0.500 kg. m 8. A soccer player extends her lower leg in a kicking motion by exerting a force with the muscle above the knee in the front of her leg. She produces an angular acceleration of 30.00 rad/s and her lower leg has a moment of inertia of 0.750 kg. m. What is the force exerted by the muscle if its effective perpendicular lever arm is 1.90 cm? (OpenStax 10.13) 1.18 103 N 9. Suppose you exert a force of 180 N tangential to a 0.280-m-radius 75.0-kg grindstone (a solid disk). (a) What torque is exerted? (b) What is the angular acceleration assuming negligible opposing friction? (c) What is the angular acceleration if there is an opposing frictional force of 20.0 N exerted 1.50 cm from the axis? (OpenStax 10.14) 50.4 N m, 17.1 rad/s, 17.0 rad/s Physics 04-05 Dynamics of Rotational Motion Dynamics of Rotational Motion Newton's Second Law for Rotation is in Moment of I = mr of a Moment of Inertia (1) measures how much an (or not start, R Use to find I = mr Unit: I= MR2 Work for rotation Kinetic Energy Conservation of Mechanical Energy PE + KE = PE, + KE Remember that the can include both and Zorch, an archenemy of Superman, decides to slow Earth's rotation to once per 28.0 h by exerting an opposing force at and parallel to the equator. Superman is not immediately concerned, because he knows Zorch can only exert a force of 4.00107 N (a little greater than a Saturn V rocket's thrust). How long must Zorch push with this force to accomplish his goal? (This period gives Superman time to devote to other villains.) wants to keep Axis Hoop about cylinder axis Axis Name: = (R + R2) | = 90 Solid cylinder (or disk) about cylinder axis Axis Solid cylinder (or disk) about central diameter 1 = MR + Me 4 12 1 = MR2 2 Axis Thin rod about axis through center 1 to length Axis 1 = Me2 12 1 = Me 3 Axis Axis 2R Solid sphere about any diameter 1 = 2MR2 5 Axis Hoop about R any diameter 1=MR2 2 I = 2MR2 3 Axis M (a2+ b) 1= 12 Thin rod about axis through one end to length FT Thin 2R spherical shell about any diameter Slab about I axis through center A solid sphere (m = 2 kg and r = 0.25 m) and a thin spherical shell (m = 2 kg and r = 0.25 m) roll down a ramp that is 0.5 m high. What is the velocity of each sphere as it reaches the bottom of the ramp?