Angular Momentum and Rotation Extended

Angular Momentum and Rotation Extended 1. A cyclist traveling at 5m/s uniformly accelerates up to 10 m/s in 2 seconds. Each tire of the bike has a 35 cm radius, and a small pebble is caught in the tread of one of them. (A) What is the angular acceleration of the pebble during those two seconds? (B) Through what angle does the pebble revolve? (C) How far does the pebble travel during that acceleration interval? (linear distance) 2. A 10 kg mass hangs on a rope wrapped around a freely rotating 2 kg cylinder of radius 10 cm. Determine the tension in the rope and the angular and tangential acceleration. 3. A solid sphere of radius R is rolling freely without slipping down an incline of angle . Determine the tangential acceleration of the sphere in terms of  and g. 4. A solid sphere of radius 37 cm is on an incline 5 meters long that makes an angle of 25 with the horizontal. Assuming the sphere starts from rest, what is its final linear speed as it reaches the bottom of the ramp. 5. The evil Dr. L is involved in a plot to de-spin the Earth using knowledge acquired from the Franklin Institute. HE plans to mount a series of surplus rockets tangentially all along the equator. Taking the planet to be a uniform sphere of radius 6.37E6 m and mass 5.98E24 kg, how much continuous thrust would the rockets need to apply to accomplish the deed in 12 hours? 6. Taking the Earth to be a uniform sphere with dimensions as given above, compute its angular momentum about its spin axis. 7. A small particle of mass 0.2 kg is being whirled in a horizontal circle at the end of a 2m long string at a constant speed of 1m/s. (A) Determine its angular momentum about its axis of rotation. (B) Suppose the string is quickly reeled in leaving the radius to be 1m. Determine the new angular velocity. 8. The moment of inertia of a spinning body about its spin axis depends on (A) angular speed, shape and mass (B) angular acceleration, mass , axis position (C) mass, shape, axis position, size (D) mass, size, shape and speed 9. Every point mass on a rigid body that is rotating around a fixed axis has (A) same linear and angular speed (B) different angular speed but same angular acceleration (C) the same angular speed and angular acceleration (D) same angular speed and same linear acceleration (E) none of these 10. A child’s top is spun with angular acceleration given as: α = 8t3 - 4t, assuming the initial conditions to be zero, a. find the angular velocity at 4 seconds. b. Find the angular acceleration at 2 seconds c. Find the angular distance at 1 second. d. Is the angular acceleration constant? 11. A hoop (MR2), a disk (1/2MR2) and a sphere (2/5MR2) are start at the top of a ramp of length 2.5m making an angle of 12˚ with the horizontal. a. Which objects gets to the bottom the fastest? b. Which object has the greatest kinetic energy at the bottom? c. Calculate the translational velocity for each object. 12. Wheel A and B (below) are connected by a belt that does not slip. The radius of wheel B is 3 times that of A. What is the ratio of the inertias Ia/Ib if the two wheels had the same a) angular momentum b) rotational kinetic energy B A 13. The pulley shown below has radius R = 20cm and a mass of 2 kg. If the object is considered to be a cylinder (I=1/2Mr2), determine the acceleration of the system, assuming the block slides down the incline. The angle the ramp makes is 25˚ with the horizontal. The coefficient of friction of between the block and ramp is μ=0.4. M1 = 18 kg M2=7kg