To solve this problem, recognize that the force due to friction must equal the centripetal force of the curve:

This will give the maximum force that the car can have in the curve without skidding. Expand the equation for centripetal force.

We are given the value for the force of friction, the mass of the car, and the radius of the curve. Using these values, we can find the velocity.

Torque is a force times the radius of the circle, given by the formula:

In this case, we are given the radius in centimeters, so be sure to convert to meters:

Use this radius and the given force to solve for the torque.

The formula for the potential energy in a spring is:

Use the given spring constant and displacement to solve for the stored energy.

We also can calculate the total number of seconds in a day.

There are seconds in one day.

Therefore we want our clock to swing a certain number of times with a period of to equal .

We know that our current clock has a certain number of swings with a period of to equal

So we have

We can calculate the current period of the pendulum using the equation

We can set up a ratio of each of these two periods to determine the missing length.

Notice that 2, pi and g are all in both the numerator and denominator and therefore fall out of the problem.

We can now solve for our missing piece.

Square both sides to get rid of the square root.

We should lengthen the pendulum by

The formula for the potential energy in a spring is:

Use the given spring constant and displacement to solve for the stored energy.

We can use the conservation of momentum to solve. Since the satellites stick together, there is only one final velocity term.

We know the masses for both satellites are equal, and the second satellite is initially stationary.

Now we need to find the velocity of the first satellite. Since the satellite is in orbit (circular motion), we need to find the tangential velocity. We can do this by finding the centripetal acceleration from the centripetal force.

Recognize that the force due to gravity of the Earth on the satellite is the same as the centripetal force acting on the satellite. That means .

Solve for for the satellite. To do this, use the law of universal gravitation.

Remember that r is the distance between the centers of the two objects. That means it will be equal to the radius of the earth PLUS the orbiting distance.

Use the given values for the masses of the objects and distance to solve for the force of gravity.

Now that we know the force, we can find the acceleration. Remember that centripetal force is Fc=m∗ac. Set our two forces equal and solve for the centripetal acceleration.

Now we can find the tangential velocity, using the equation for centripetal acceleration. Again, remember that the radius is equal to the sum of the radius of the Earth and the height of the satellite!

This value is the tangential velocity, or the initial velocity of the first satellite. We can plug this into the equation for conversation of momentum to solve for the final velocity of the two satellites.

The impulse is equal to the change in momentum. The change in momentum is equal to the mass times the change in velocity.

The yellow ball rebounds higher and therefore has a higher velocity after the rebound. Since it has a higher velocity after the collision, the overall change in momentum is greater. Therefore since the change in momentum is greater, the impulse is higher.

We know that the work-kinetic energy theorem states that the work done is equal to the change of kinetic energy. In rotational terms this means that

In this case the initial angular velocity is .

We can convert our final angular velocity to radians per second.

We also can calculate the moment of inertia of the merry-go-round assuming that it is a uniform solid disk.

We can put this into our work equation now.

We know that the work-kinetic energy theorem states that the work done is equal to the change of kinetic energy. In rotational terms this means that

In this case the initial angular velocity is .

We can convert our final angular velocity to radians per second.

We also can calculate the moment of inertia of the merry-go-round assuming that it is a uniform solid disk.

We can put this into our work equation now.

For this problem, we'll use Newton's third law, which states that for every force there will be another force equal in magnitude, but opposite in direction.

This means that the force of the first skater on the second will be equal in magnitude, but opposite in direction:

Use Newton's second law to expand this equation.

We are given the mass of each skater and the acceleration of the first. Using these values, we can solve for the acceleration of the second.

From here, we need to isolate the acceleration of the second skater.

Notice that the acceleration of the second skater is negative. Since she is moving in the opposite direction of the first skater, one acceleration will be positive while the other will be negative as acceleration is a vector.