The force required to make an object of mass m, travelling with velocity v, turn in a circle of radius r is

Analysis

The question asks for the force required to keep an object of mass $m$ moving in a circular path of radius $r$ with a constant velocity $v$. This is a fundamental concept in circular motion known as centripetal force.

Concept

When an object moves in a circular path, it is constantly changing its direction. Even if the speed ($v$) remains constant, the change in direction means the object is accelerating. This acceleration is directed toward the center of the circle and is called centripetal acceleration ($a_c$).

The formula for centripetal acceleration is:

According to Newton's Second Law of Motion ($F = ma$), the force required to produce this acceleration is the product of the mass and the centripetal acceleration.

Solution

1. Identify the variables: • Mass = $m$ • Velocity = $v$ • Radius = $r$
2. Apply Newton's Second Law: • $F = m \times a_c$
3. Substitute the centripetal acceleration formula: • $F = m \times \left(\frac{v^2}{r}\right)$
4. Simplify: • $F = \frac{mv^2}{r}$

This force is what we call the centripetal force. It is the "center-seeking" force that prevents the object from flying off in a straight line (tangent to the circle).

Conclusion

Comparing our result to the given options, the correct expression is $\frac{mv^2}{r}$.

Exam Tip: Always remember that "centripetal" means "center-seeking." If you ever forget the formula, remember that the force must increase if the speed increases (square of velocity) and must decrease if the radius of the turn increases (inversely proportional to $r$). This logic helps you eliminate options like C and D quickly!