Question

A body of mass 2 kg is moving under the influence of a central force whose potential energy is given by U (r) = 2r3 Joule. If the body is moving in a circular orbit of 5m,then find its energy.

Answer 1

Answer on Question#56931 - Physics - Mechanics - Relativity

A body of mass $m = 2 \, \text{kg}$ is moving under the influence of a central force whose potential energy is given by $U(r) = 2r^3$ Joule. If the body is moving in a circular orbit of $R = 5 \, \text{m}$, then find its energy.

Solution:

The force of potential field

$$F(r) = -\nabla U(r) = -6r^2 \frac{r}{r}$$

Since the body is moving in a circular orbit, the centripetal force is given by force $F(r)$:

$$m \frac{v^2}{r} = |F(r)| = 6r^2,$$

Where $v$ – is the speed of the body.

Thus

$$mv^2 = 6r^3 = 3U(r)$$

Therefore, the kinetic energy $E_k$ is given by

$$E_k = \frac{mv^2}{2} = \frac{3U(r)}{2}$$

The total energy is given by the sum of the potential energy and kinetic energy:

$$E(r) = U(r) + E_k = U(r) + \frac{3}{2} U(r) = \frac{5}{2} U(r)$$

Thus

$$E(R) = \frac{5}{2} U(R) = \frac{5}{2} \cdot 2 \cdot 5^3 \, \text{J} = 625 \, \text{J}$$

Answer: 625 J.

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