In order to find the radius of the circle that corresponds to the maximum tension the string can experience without breaking, we must rearrange the centripetal force equation for objects moving in a circle:

To solve this problem, it is important to remember the signs of each force acting on the coaster.  Since we are inside the loop at the top, both the weight and the normal force point toward to center of the circular path, and are thus positive.

Next, we note that the normal force is what a rider "feels" during their ride on the rollercoaster.  We want the speed at which a rider would feel three times as heavy, or three times their weight.

Centripetal acceleration  can be given as:

, where  is the linear velocity,  is the radius of the circle. 

To determine linear velocity, we need to determine how quickly the wheel spins.

Firstly, let's convert from rpm to Hertz. 

The wheel makes 50 full revolutions in one second. To determine linear velocity,

, where  is frequency, which we just determined and  is circumference of the tire. 

By the definition of the circumference of a circle,

Therefore, 

The formula for centripetal acceleration is given by:

 is the centripetal acceleration,  is the velocity and  is the radius of the circular path. We substitute in our known values and then solve for alpha which will give us the centripetal acceleration of the ferris wheel.

Be aware that the radius is half of the diameter, so  will be 5 meters.

This will give us a centripetal acceleration of 

The tension and gravity both point down toward the center of the circle. The net force on the ball is the vector sum of these two: 

For an object in circular motion, the net force must be equal to:

In this case, the force of gravity is helping the tension provide this force, so the tension is not as great as it would be if it were supplying the entire centripetal force.

Substitute and solve.

For the car going around a flat turn, the force of friction is the only force keeping the car in a circular path. This lets us equate the centripetal force to the force of friction.

Substitute and solve for the coefficient of friction.

Since the tires are rotating but not slipping on the surface, this is the coefficient of static friction, .

When a car is going around a flat turn, the only force keeping the car going along a circular path is the force of friction, which is equal to the centripetal force. We do not need the mass of the car to calculate the maximum speed, as we will show below.

Rearrange these equations and solve for the velocity.

For a car going around a banked turn with no friction, a component of the normal force  is what provides the centripetal force. The radius of the curve can be calculated without knowing the mass of the car.

Centripital acceleration is

Force is defined as

Using subsitution

The rock has a frequency of .

Each circle it makes has a circumference of , which is equivalent to .

That means it travel  in , which is

Plugging in our values:

We use the formula for centripetal acceleration:

Where  is the centripetal acceleration,  is the velocity  and  is the radius (1m).

Plug in these known values and solve.