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Gravitational Potential

A thin uniform annular disc (see figure) of mass M has outer r

Question

Medium

Solving time: 3 mins

A thin uniform annular disc (see figure) of mass $M$ has outer radius $4 R$ and inner radius $3 R$ . The work required to take a unit mass from point $P$ on its axis to infinity is

\frac{2 GM}{7 R} (42 - 5)

- \frac{2 GM}{7 R} (42 - 5)

\frac{GM}{4 R}

\frac{2 GM}{5 R} (2 - 1)

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Updated on: Apr 29, 2023

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A thin uniform annular disc (see figure) of mass

M

has outer radius

4 R

and inner radius

3 R

. The work required to take a unit mass from point

P

on its axis to infinity is

8 mins ago

Discuss this question LIVE

8 mins ago

Text solutionVerified

W = Δ U = U_{f} - U_{i} = U_{\infty} - U_{p}

= - U_{p} = - m V_{p}

= - V_{p} (as m = 1)

Potential at point

P

will be obtained by in integration as given below. Let

d M

be the mass of small rings as shown
Solution Image

d M d V_{P} ∴ W = \frac{M}{π ( 4 R ) ^{2} - π ( 3 R ) ^{2}} (2 π r) d r = \frac{2 M r d r}{7 R ^{2}} = - \frac{G \cdot d M}{16 R ^{2} + r ^{2}} = - \frac{2 GM}{7 R ^{2}} \int_{3 R}^{4 R} \frac{r}{16 R ^{2} + r ^{2}} \cdot d r = - \frac{2 GM}{7 R} (42 - 5) = + \frac{2 GM}{7 R} (42 - 5)

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Question Text	A thin uniform annular disc (see figure) of mass $M$ has outer radius $4 R$ and inner radius $3 R$ . The work required to take a unit mass from point $P$ on its axis to infinity is
Updated On	Apr 29, 2023
Topic	Gravitation
Subject	Physics
Class	Class 11
Answer Type	Text solution:1 Video solution: 6
Upvotes	570
Avg. Video Duration	6 min

A thin uniform annular disc (see figure) of mass $M$ has outer radius $4 R$ and inner radius $3 R$ . The work required to take a unit mass from point $P$ on its axis to infinity is

Text solutionVerified

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A thin uniform annular disc (see figure) of mass M has outer radius 4R and inner radius 3R. The work required to take a unit mass from point P on its axis to infinity is

Text solutionVerified

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A thin uniform annular disc (see figure) of mass $M$ has outer radius $4 R$ and inner radius $3 R$ . The work required to take a unit mass from point $P$ on its axis to infinity is