Instructions:
- The marks are indicated in the right-hand margin.
- There are NINE questions in this paper.
- Attempt FIVE questions in all.
- Question No. 1 is compulsory.
- Symbols used (if any) have their usual meanings
Q.1 Choose the correct answer of the following (Any seven question
only):
The moment of inertia of an arbitrarily shaped rigid body is a
The curl of a gradient is
In a lightly damped oscillator the frequency
As opposed to a standing bicycle a moving bicycle does not fall easily because
The Foucault pendulum undergoes precession because of
Which of the following is NOT Kepler's law
Which of the following is not true for conservative force
The Newton's law gives the relationship between force, mass and
Velocity or energy resonance in the forced oscillator take place when
Where $ \omega $ is frequency of force, $ \omega_0 $ is natural frequency of the oscillator and $ 2b = \frac{B}{m} $ is damping coefficient
A spinning top is an example of
Q.2 Solve both questions :
State the definition of scalar and vector quantities. Explain how they transform under rotation.
Under an anti-clockwise rotation by an angle $ \theta $ of a 2d Cartesian coordinate system find out the transformation rules for the components of an arbitrary vector $ \vec{A} $. Show that the scalar product $ \vec{A} \cdot \vec{B} $ is invariant under rotation of the coordinate system.
Q.3 Solve both questions :
Define inertial and non-inertial frames. Explain the meaning of fictitious force
Derive an expression for acceleration in a rotating frame and obtain the centripetal and Coriolis terms in the acceleration. Explain briefly how the Coriolis forces are useful in explaining the weather system
Q.4 Write short technical notes on:
Q.5 Solve all parts :
Show that the conservative force acting on a particle is always perpendicular to the equipotential surface.
Prove that the work done by a force which is the gradient of a potential ($ \vec{F} = -\nabla U $) is path independent.
The potential function is given as $ U = -2x - z^2 $. Find the work done by the resulting force in moving a particle from the origin to a point (2,2,2) along a straight line.
Q.6 Solve both questions :
Explain overdamped, underdamped and critically damped harmonic motion.
Write an equation for a forced harmonic oscillator and obtain the conditions of resonance.
Q.7 Solve both questions :
Derive the moment of inertia tensor of a rigid body and explain the concept of principal axes.
A uniform solid disc of mass $ M $ and radius $ R $ having moment of inertia $ MR^2/2 $ about an axis passing through its centre has initial uniform angular velocity. A constant friction force of magnitude $ f $ is then applied on the rim in tangential direction so that the disc finally comes to a stop in time $ T $. Find the magnitude of the friction force.
Q.8 Solve both questions :
Derive the conditions for elliptic, hyperbolic and parabolic orbits in central force motion.
Show that the total angular momentum of a solid body can be written as the sum of the angular momentum of the centre of mass and the angular momentum about the centre of mass
Q.9 Solve both questions :
Derive Euler's equations for the rotational motion of a rigid body and explain its importance.
Prove that the angular momentum is conserved in a central force motion and that the orbit of the particle is always planar.
Instructions:
- The marks are indicated in the right-hand margin.
- There are NINE questions in this paper.
- Attempt FIVE questions in all.
- Question No. 1 is compulsory.
- Symbols used (if any) have their usual meanings
Q.1 Choose the correct answer of the following (Any seven question
only):
The moment of inertia of an arbitrarily shaped rigid body is a
The curl of a gradient is
In a lightly damped oscillator the frequency
As opposed to a standing bicycle a moving bicycle does not fall easily because
The Foucault pendulum undergoes precession because of
Which of the following is NOT Kepler's law
Which of the following is not true for conservative force
The Newton's law gives the relationship between force, mass and
Velocity or energy resonance in the forced oscillator take place when
Where $ \omega $ is frequency of force, $ \omega_0 $ is natural frequency of the oscillator and $ 2b = \frac{B}{m} $ is damping coefficient
A spinning top is an example of
Q.2 Solve both questions :
State the definition of scalar and vector quantities. Explain how they transform under rotation.
Under an anti-clockwise rotation by an angle $ \theta $ of a 2d Cartesian coordinate system find out the transformation rules for the components of an arbitrary vector $ \vec{A} $. Show that the scalar product $ \vec{A} \cdot \vec{B} $ is invariant under rotation of the coordinate system.
Q.3 Solve both questions :
Define inertial and non-inertial frames. Explain the meaning of fictitious force
Derive an expression for acceleration in a rotating frame and obtain the centripetal and Coriolis terms in the acceleration. Explain briefly how the Coriolis forces are useful in explaining the weather system
Q.4 Write short technical notes on:
Q.5 Solve all parts :
Show that the conservative force acting on a particle is always perpendicular to the equipotential surface.
Prove that the work done by a force which is the gradient of a potential ($ \vec{F} = -\nabla U $) is path independent.
The potential function is given as $ U = -2x - z^2 $. Find the work done by the resulting force in moving a particle from the origin to a point (2,2,2) along a straight line.
Q.6 Solve both questions :
Explain overdamped, underdamped and critically damped harmonic motion.
Write an equation for a forced harmonic oscillator and obtain the conditions of resonance.
Q.7 Solve both questions :
Derive the moment of inertia tensor of a rigid body and explain the concept of principal axes.
A uniform solid disc of mass $ M $ and radius $ R $ having moment of inertia $ MR^2/2 $ about an axis passing through its centre has initial uniform angular velocity. A constant friction force of magnitude $ f $ is then applied on the rim in tangential direction so that the disc finally comes to a stop in time $ T $. Find the magnitude of the friction force.
Q.8 Solve both questions :
Derive the conditions for elliptic, hyperbolic and parabolic orbits in central force motion.
Show that the total angular momentum of a solid body can be written as the sum of the angular momentum of the centre of mass and the angular momentum about the centre of mass
Q.9 Solve both questions :
Derive Euler's equations for the rotational motion of a rigid body and explain its importance.
Prove that the angular momentum is conserved in a central force motion and that the orbit of the particle is always planar.