Describe, qualitatively, motion in a circular path due to a force perpendicular to the motion as: (a) speed increases if force increases, with mass and radius constant (b) radius decreases if force increases, with mass and speed constant (c) an incre

1.5.1 Effects of Forces

Objective

Describe, qualitatively, motion in a circular path due to a force perpendicular to the motion as:

1. Speed increases if the force increases, with mass and radius constant.
2. Radius decreases if the force increases, with mass and speed constant.
3. An increased mass requires an increased force to keep speed and radius constant.

Key relationship

The magnitude of the centripetal force required to keep an object moving in a circle of radius r with speed v is

\$F = \frac{mv^{2}}{r}\$

where m is the mass of the object.

Qualitative analysis

(a) Increasing the force while keeping mass and radius constant

If m and r are fixed, the equation can be rearranged to

\$v = \sqrt{\frac{Fr}{m}}\$

Thus, a larger force F leads to a larger speed v. The object moves faster around the same circular path.

(b) Increasing the force while keeping mass and speed constant

With m and v fixed, the equation becomes

\$r = \frac{mv^{2}}{F}\$

Therefore, a larger force produces a smaller radius. The path contracts, pulling the object closer to the centre.

(c) Increasing the mass while keeping speed and radius constant

When v and r are unchanged, the required force is

\$F = \frac{mv^{2}}{r}\$

Increasing the mass directly increases the needed centripetal force. More force must be supplied to maintain the same circular motion.

Summary table