Notes : Shell Theorem and Gravitation Shielding - FAQ and Quiz - Class 11 Physics
Shell Theorem
The Shell Theorem describes the gravitational effect of a uniform hollow spherical shell on a point mass placed either outside or inside the shell.
(i) Point Mass Outside a Uniform Spherical Shell
Statement:
The force of attraction between a hollow spherical shell of uniform density and a point mass situated outside it is just as if the entire mass of the shell were concentrated at the centre of the shell.
The gravitational force is given by:
\[ F=\frac{GMm}{r^2} \]
where,
- G = Universal gravitational constant
- M = Mass of the shell
- m = Mass of the particle
- r = Distance of the particle from the centre of the shell
(ii) Point Mass Inside a Uniform Spherical Shell
Statement:
The force of attraction due to a hollow spherical shell of uniform density on a point mass situated inside it is zero.
\[ F = 0 \]
Summary of Shell Theorem
- Inside the shell (r < R): \(F = 0\)
- Outside the shell (r > R): \(\displaystyle F=\frac{GMm}{r^2}\)
where R is the radius of the spherical shell.
Application to Earth
Since the Earth is approximately spherical, the gravitational force on a body of mass m near the Earth's surface is:
\[ F=\frac{GMm}{R^2} \]
Since,
\[ F = mg \]
therefore,
\[ g=\frac{GM}{R^2} \]
Gravitational Shielding
A body placed inside a hollow spherical shell does not experience gravitational force due to the shell itself. However, it still experiences gravitational force due to bodies outside the shell.
Therefore, gravitational shielding is not possible.
FAQs : Frequently Asked Questions
1. What does the shell theorem state?
It states that the gravitational force inside a uniform spherical shell is zero, while outside the shell it acts as if all its mass were concentrated at the centre.
2. What is the gravitational force on a particle inside a hollow spherical shell?
The gravitational force is zero.
\[ F = 0 \]
3. How does a spherical shell behave for a point mass outside it?
It behaves as if the entire mass of the shell were concentrated at its centre.
\[ F=\frac{GMm}{r^2} \]
4. Is gravitational shielding possible?
No. A hollow shell cannot shield a body from gravitational forces due to external objects.
5. Can the shell theorem be applied to planets like Earth?
Yes. For points outside the Earth, the Earth behaves approximately as a point mass located at its centre.
Quiz – Shell Theorem
1. The gravitational force on a particle placed inside a uniform spherical shell is:
A. Maximum
B. Equal to mg
C. Zero
D. Infinite
Answer: C. Zero
2. A spherical shell attracts an external particle as if:
A. Its mass is uniformly distributed over the surface
B. Its mass is concentrated at the centre
C. Its mass is concentrated at the point of contact
D. No force acts on the particle
Answer: B. Its mass is concentrated at the centre
3. The gravitational force on a mass m outside a shell of mass M is given by:
A. \(\frac{GMm}{r}\)
B. \(\frac{GMm}{r^2}\)
C. \(\frac{GMm}{r^3}\)
D. \(GMmr^2\)
Answer: B. \(\frac{GMm}{r^2}\)
4. Gravitational shielding is:
A. Possible using a thick shell
B. Possible using a conducting shell
C. Possible inside Earth
D. Not possible
Answer: D. Not possible
5. The shell theorem is applicable to:
A. Only cubical bodies
B. Only cylindrical bodies
C. Spherically symmetric mass distributions
D. All irregular bodies
Answer: C. Spherically symmetric mass distributions
Key Points
- Inside a uniform spherical shell, gravitational force is zero.
- Outside the shell, it behaves as if all its mass were concentrated at its centre.
- The theorem is valid for spherically symmetric mass distributions.
- Gravitational shielding is not possible.
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