Question

The force to keep an object turning in a circle is given by F = m×V^2/d. You swing a ball on a string. The same force is required to swing the ball three times as fast (3V) at distance d as is required to swing it at speed V at ____ the distance.

Answer 1

When the speed of the ball is tripled, the centripetal force required is the same as if the ball were swung at its original speed but at nine times the distance. This follows from the centripetal force formula F = m × V^2 / d, where velocity is squared.

Step-by-step explanation:

The question is dealing with the concept of centripetal force, which is the force needed to keep an object moving in a circular path. Specifically, when you swing a ball three times faster at distance d, the force required for this is equal to the force required to swing the ball at speed V at nine times the distance. This conclusion comes from the formula for centripetal force, which can be written as F = m × V^2 / r, where F is the centripetal force, m is the mass of the ball, V is the velocity of the ball, and r or d in this case is the distance from the center of the circle to the ball.

To answer the question: If you increase the speed to 3V, the force required can be calculated as F' = m × (3V)^2 / d = 9 × m × V^2 / d. This is the same force that would be required for swinging the ball at speed V at a distance r' = 9 × d, since F' = m × V^2 / r', due to the square relationship between velocity and the centripetal force.