Question

Physics Question:

A wheel has a radius of .5 m and a moment of inertia I = 3kg m^2. It is rotating at 20 rev/s about an ais through its center when a stick, acting as a brake, is applied to the outer edge. It is brought to rest in 30 seconds

a.) Find the angular acceleration of the wheel

b.) How many revolutions does it make before coming to rest?

c.) How much kinetic energy does it lose in coming to rest?

d.) How much force did the stick exert on the wheel?

e.) Find the linear distance covered by a speck of dust, located halfway between the center of the wheel and the outer edge, during the time the wheel was coming to rest

Answer 1

This problem uses rotational kinematics as the wheel slows to a stop.

a)

Step 1) To find the angular acceleration of the wheel, use the equation , where is the final angular speed, is the intial angular speed, is the angular acceleration, and t is the time. The intial angular speed is given to be . Convert this into radians per second by multiplying by the conversion factor .

Step 2) Plug in , , and into to solve for the angular acceleration.

The angular acceleration is -4.19 radians per second squared. The negative sign indicates that it is slowing down.

b)

Step 3) To find how many revolutions the wheel makes before it comes to a rest, use the equation where is the initial angular position and is the final angular position. Plug in , , and and solve for .

Step 4) Convert this angle into revolutions by multiplying by the conversion factor .

The wheel undergoes 300 revolutions as it comes to a stop.

c)

Step 5) To figure out how much kinetic energy it loses while coming to a rest, realize that the formula for rotational kinetic energy K is where I is the moment of inertia of the wheel. When the wheel comes to a stop, it has lost all of its initial kinetic energy. Plug in and into .

The kinetic energy lost is 23701 Joules.

d)

Step 6) To figure out how much force was exerted by the stick on the wheel in order to make it come to a stop, use Newton's 2nd law for rotational dynamics which states that the sum of the torques acting on a body is equal to the moment of inertia of the body times its angular acceleration, or . Plug in and to find the net torque applied to the wheel.

The negative sign just tells us what direction the torque is acting in.

Step 7) Now use the formula for the torque given by , where r is the radius of the wheel and F is the force of the stick on the wheel causing the torque. Since the force and the wheel are perpendicular, the cross product simplifies to . Plug in and and solve for F.

The stick applies a force with magnitude 25.14 N.

e)

Step 8) To find the linear distance covered by a speck of dust halfway between the center of the wheel at the rim while the wheel comes to a stop, use the arc length formula where s is the linear distance the speck travels, r is the distance the speck is from the center, and is the angular displacement of the wheel as it comes to a stop. Plug in and and solve for s.

The speck travels 471.4 meters as the wheel comes to a stop.