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Lesson 3, Topic 4
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# Gravitational Potential (V)

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Whenever a body is lifted against the force of gravity, work is done on the body. Likewise, when a body is lifted from earth’s surface to a new position above the earth’s surface, the work is stored in the body as potential energy. E.g. if a body is moved from a point to another point r from the earth’s surface, the work done = force x displacement

= $$\frac{GMM} {r^2} \scriptsize \; \times \; r$$

= $$\frac{GMM} {r}$$

V = $$\frac{GMM} {r}$$

Where M- Mass of the earth, m-mass of body moved against the earth

r- Distance of the mass from earth’s surface

A man outside the earth’s gravitational field is at infinity ( ∞) and the gravitational potential energy is zero at infinity.

When a mass at infinity is moved to a new location on the earth’s surface, work is done, and v = $$\scriptsize v_ {\infty} – v_r$$

$$\scriptsize v_r$$ = potential at new location

$$\scriptsize v_{\infty}$$ =potential at infinity

$$\frac{GMM} {r} \scriptsize = 0 – v_r$$ $$\scriptsize v_r = \normalsize \frac{GMM} {r}$$

Work done per unit mass is called gravitational potential, = $$\frac{GM} {r}$$

Therefore, gravitational potential is the work done in moving a unit mass (kg) from infinity to other point in the earth’s gravitational field.

The negative sign (-) shows that potential at vr is less than potential at infinity, $$\scriptsize v_{\infty}$$

A rocket of mass 1000kg is resting on the earth’s surface. If the earth’s radius is 6.4 x 10 6  and mass is 6.0 x 10 24, calculate:

• P.e of the rocket on earth’s surface
• P.e of the rocket at height 1.0  x 10 6 m
• Gain in P.e of the rocket position in the new position

Vy = $$\frac{GMm} {r}$$

= $$\frac{6.67 \times 10^{-11} \; \times \; 1000 \; \times \; 6.0 \times 10^{24}} {6.4 \times 10^{6}}$$

= 6.523 x 10 10

Vr = $$\scriptsize – v_0 \normalsize \frac{r}{r+h}$$

= $$\scriptsize – 6.523 \times 10^{10} \; \times \; \normalsize \frac{6.4 \times 10^{6}}{6.0 \times 10^{-24} + 1.0 \times 10^{6}}$$

= -5.408 x 10 10 J

Gain in P.e = Vr – Vo

= -5.408 x 10 10 – (-6.523 x 10 10)

=8.45 10 9J

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