Question

A ball of mass M at the end of a string is swinging in a horizontal circular path of radius R at a constant speed V. Which combination of changes would require the greatest increase in the centripetal force acting on the ball?

1) doubling V and R
2) Doubling V and halving R
3) halving V and doubling R
4) halving V and hlaving R

Answer 1

The greatest increase in centripetal force occurs when the velocity is doubled and the radius is halved, producing an eightfold increase in the force required. Option 2.

Step-by-step explanation:

To determine which change requires the greatest increase in centripetal force, we have to consider the formula for centripetal force, which is Fc = mV2/R, where Fc is the centripetal force, m is the mass, V is the velocity, and R is the radius of the circular path.

When making changes to the velocity V and the radius R, we see different impacts on Fc:

Doubling V and R leads to the new force F' = m(2V)2/(2R) = 2mV2/R, which is double the original force.

Doubling V and halving R leads to the new force F' = m(2V)2/(0.5R) = 8mV2/R, which is eight times the original force.

Halving V and doubling R leads to the new force F' = m(0.5V)2/(2R) = 0.125mV2/R, which is one-eighth the original force.

Halving V and halving R maintains the original force F' = m(0.5V)2/(0.5R) = mV2/R.

From these calculations, the greatest increase in centripetal force occurs when the velocity is doubled and the radius is halved, resulting in an eightfold increase in the force required. Option 2.