Chapter 3, Problem 10MDP

1. 10. Show that for circular motion, force = mass * velocity squared/radius.

To determine

Show that the force of the circular motion is, $F=\frac{m{v}^2}{R}$.

Answer to Problem 10MDP

The force of the circular motion is proved as $\underset{\_}{F=\frac{m{v}^2}{R}}$.

Explanation of Solution

Circular motion is defined as the movement of an object along a circular path. When an object is moved at constant speed, the velocity is changed due to its direction not because of its magnitude. This changing velocity shows that the object is accelerating. To obtain this accelerating there should be a force which is called as centripetal force.

The derivation of the force of the circular motion is as follows,

Consider a ball moving with the constant speed $v$ and radius $R$ as shown in Figure 1.

To derive the acceleration consider the initial velocity at $A$ and the final velocity at $B$.

Now modify Figure 1 as shown in Figure 2.

Use the similar triangles in Figure2, therefore it becomes as follows:

The acceleration is the rate of change of velocity.

$a=\frac{\Delta v}{t}$ (1)

Using similar triangle theorem,

$\frac{\Delta v}{v}=\frac{s}{R}$ (2)

The arc with the chord is,

$s=vt$ (3)

Rearrange equation (3) to find $v$.

$v=\frac{s}{t}$ (4)

Rearrange equation (2) to find $\Delta v$.

$\Delta v=v\left(\frac{s}{R}\right)$ . (5)

Substitute equation (5) in equation (1) to obtain the expression of acceleration in terms of $v$ and s.

$a=\frac{v\left(\frac{s}{R}\right)}{t}$

$a=\frac{vs}{Rt}$ (6)

Substitute $v$ for $\frac{s}{t}$ in equation (6) to find acceleration.

$a=\frac{v\left(v\right)}{R}$

$a=\frac{v^2}{R}$ (7)

Write the expression for force.

$F=ma$ . (8)

Substitute equation (7) in (8) to obtain the expression force of the circular motion.

$F=\frac{m{v}^2}{R}$ (9)

Here,

$m$ is the mass,

$v$ is the velocity,

$R$ is the radius.

Therefore, the equation (9) shows the force of the circular motion.

Conclusion:

Thus, the force of the circular motion is proved as $\underset{\_}{F=\frac{m{v}^2}{R}}$.