Through conservation of angular momentum, as moment of inertia decreases, the angular velocity increases. Moment of inertia is dependent on the distribution of the spinning body's mass away from the center of mass—as the skater brings his arms in, he lowers his moment of inertia, thus increasing his angular velocity. If he raises his arms completely vertically from this point, it does not change the radius of mass distributions (from the center of his body), thus maintaining his angular velocity.

By the conservation of angular momentum, the angular momentum , is equal to the product of the mass, angular velocity, and radius (or length of the rope in this case). The equation relating these terms is:

Here,  is the initial mass,  is the initial angular velocity, and  is the length of the rope, which remains constant. Angular momentum must be conserved, thus: 

Substitute.

We are given that  and  is the final velocity. Plug in and solve.

Since the platform is frictionless, as the child walks, angular momentum is conserved. However, since there is an  term in the kinetic energy expression, as  decreased due to the increase of inertia, it affects the energy more, and it decreases. From the reference frame of the disk, the child feels an outward directed force, and thus is doing negative work. This is the same principle that allows ice skaters to increase their spinning speed.

For the first disk, we have the information below.

For the second disk, we have the information below.

(The minus sign indicates counterclockwise while the positive indicates clockwise.)

To do this problem, we use conservation of angular momentum. Before the disks are put in contact, the initial total angular momentum is given by the equation below.

It is just the sum of the angular momentums of each disk. When the disks are combined together, then final angular momentum can be found by the following equation.

Set this equal to the initial angular momentum.

Simplify and solve for .

The equation for conservation of angular momentum is:

This means that the angular momentum of the object at the two points in its orbit must be the same. Since the mass of the debris does not change, this gives us an equality of the product of the velocity and distance of the debris at any two points of its orbit:

Plug in known values.

There is disagreement between units; the velocities are given in both  and . Glance down at the answer choices and note that they are all lengths in kilometers.

Solve for : 

Remember the torque equation:

Exerting force on the wrench at an angle less perpendicular to the wrench will reduce and thus reduce the torque.

The other answer options will all increase the torque being applied, making the twisting motion easier.

The ball is experiencing centripetal force so that it can travel in a circular path. This centripetal force is written as the equation below.

Remember that centripetal acceleration is given by the following equation.

Since the centripetal force is coming from the tension of the string, set the tension force equal to the centripetal force.

Since we're trying to find the speed of the ball, we solve for v.

We know the following information from the question.

We can use this information in our equation to solve for the speed of the ball.

The correct answer is "toward the center of the circle." Newton's second law tells us that the direction of the net force will be the same as the direction of the acceleration of the object.

In uniform circular motion, the object accelerates towards the center of the circle (centripetal acceleration); the net force acts in the same direction.

The force of friction is what keeps the car in circular motion, preventing it from flying off the track. In other words, the frictional force will be equal to the centripetal force.

We can cancel mass from either side of the equation and rearrange to solve for the coefficient of friction:

We can use our given values to solve:

The time for one full revolution can be calculated simply by manipulating the defintion of velocity, where the distance is just the circumference of the circlular path. The time it takes is modeled by the following equation: 



Use the given radius and velocity to solve for the time per revolution: