Problem Set 6: Torque and Rotational Dynamics
Six problems on the whole of Unit 5, one for each topic, out of 100 points. Ask Socrates under any part.
Equationsω=ω0+αt θ=θ0+ω0t+½αt2 v=rω aT=rα τ=rF sinθ I=Σmiri2 I′=Icm+Md2 Στ=0 α=Στ/I a=Rα
Problem 1
A potter’s wheel starts from rest and reaches an angular velocity of 12.0 rad/s in 5.00 s, turning with a steady angular acceleration.
(a) Find the angular acceleration of the wheel. 4 pts
(b) Find the angle the wheel turns through in those 5.00 s, in radians and in full turns. 4 pts
(c) On the grid, to a stated scale, sketch the angular velocity of the wheel against time for the 5.00 s. 4 pts
(d) The potter then lets go and the wheel slows at a steady 0.800 rad/s2 until it stops. Find how long the slowing takes and the total angle turned from the start, and explain which equation you used at each step and what made it the one to reach for. 5 pts
Problem 2
A motor pulley of radius 0.0600 m drives a belt that runs around a grinding wheel of radius 0.180 m. The belt does not slip. The motor pulley turns at 90.0 rad/s.
(a) Find the speed of the belt. 4 pts
(b) Find the angular velocity of the grinding wheel. 4 pts
(c) On the grid, to a stated scale, sketch the angular velocity of the grinding wheel against the angular velocity of the motor pulley, as the motor runs up from rest to 90.0 rad/s. 4 pts
(d) The motor is switched off and the pulley slows steadily to rest in 4.00 s. Find the angular acceleration of each wheel, and explain how the two can differ while the belt has one speed everywhere along it. 5 pts
Problem 3
A uniform bar turns freely about a pivot at its center. A force of 12.0 N is applied straight down at a point 0.400 m to the left of the pivot, and a force of 9.00 N is applied straight down 0.500 m to the right of it.
(a) State your sign convention, then find the torque of each force about the pivot. 4 pts
(b) Find the net torque, and state which way the bar begins to turn. 4 pts
(c) On the grid, to a stated scale, sketch the net torque against the distance of the 9.00 N force from the pivot, out to 0.800 m. 4 pts
(d) Find where the 9.00 N force has to act for the net torque to be zero, and explain why moving the 12.0 N force in by the same distance instead would not have put the bar in balance. 5 pts
Problem 4
Three small spheres are fixed to a rod 1.20 m long whose own mass is small enough to leave out: 2.00 kg at the left end, 3.00 kg at the middle, and 1.00 kg at the right end.
(a) Find the rotational inertia about an axis through the left end, perpendicular to the rod. 4 pts
(b) Find the rotational inertia about an axis through the middle sphere. 4 pts
(c) On the grid, to a stated scale, sketch the rotational inertia against the position of the axis along the rod, from end to end. 4 pts
(d) Find the position of the axis that gives the smallest rotational inertia and the value there, and explain why that position is not the middle of the rod. 5 pts
Problem 5
A uniform plank of mass 12.0 kg and length 3.00 m rests on a pivot 1.00 m from its left end. A child of mass 25.0 kg sits on the left end.
(a) Find the torque of the weight of the plank about the pivot. 4 pts
(b) Find the torque of the weight of the child about the pivot. 4 pts
(c) On the grid, to a stated scale, sketch the net torque against the position of a 15.0 kg block placed on the plank, measured from the pivot toward the right end, out to 2.00 m. 4 pts
(d) Find where the 15.0 kg block has to sit for the plank to balance, and explain why the force from the pivot never appears in your torque equation. 4 pts
Problem 6
A block of mass 2.00 kg hangs from a light string wrapped around a pulley of radius 0.150 m and rotational inertia 0.0320 kg·m2. The string does not slip and the pulley turns on a frictionless axle.
(a) Draw a force diagram for the block and one for the pulley, then write Newton’s second law for the block, the rotational form for the pulley, and the line that connects the two accelerations. 4 pts
(b) Find the acceleration of the block. 4 pts
(c) On the grid, to a stated scale, sketch the acceleration of the block against the rotational inertia of the pulley, out to 0.100 kg·m2. 4 pts
(d) Find the tension in the string, and explain why it is not equal to the weight of the block. 4 pts