Physics (Mechanics & Mechanics of Solid) - B.Tech. 1st Semester Examination, 2023

The moment of inertia of an arbitrarily shaped rigid body is a

1. Scalar
2. Vector
3. Tensor
4. Dimensionless number

The curl of a gradient is

1. Zero
2. Non-zero
3. Undefined
4. Infinite

In a lightly damped oscillator the frequency

1. Depends on the square of amplitude
2. Does not depend on the amplitude
3. Depends on the damping constant
4. Does not depend on the damping

As opposed to a standing bicycle a moving bicycle does not fall easily because

1. The cyclist keeps moving slightly left and right to maintain the C.G in the plane of the bicycle
2. The cycle is balanced by air resistance
3. The angular momentum in the wheels stabilizes the cycle
4. The cyclist leans forward to lower the CG

The Foucault pendulum undergoes precession because of

1. Gravitational force of the earth
2. Coriolis force on the earth
3. Magnetic field of the earth
4. The orbiting of the moon around the earth

Which of the following is NOT Kepler's law

1. Each planet moves in an ellipse with the Sun at one focus *
2. Planet's period of revolution \( T \) is related to the major axis of the ellipse \( A \) by \( T^2 = kA^3 \)
3. Planet's period of revolution \( T \) is related to the major axis of the ellipse \( A \) by \( T^3 = kA^2 \)
4. The radius vector from the Sun to the planet sweeps equal areas in equal times

Which of the following is not true for conservative force

1. Curl of the force is zero
2. Gravitational force is conservative in nature
3. Work done is independent of the path
4. Curl of the force is non zero

The Newton's law gives the relationship between force, mass and

1. Time
2. Velocity
3. Acceleration
4. Impulse

Velocity or energy resonance in the forced oscillator take place when

1. \( \omega = \sqrt{\omega_0^2 - 2b^2} \)
2. \( \omega = \omega_0 \)
3. \( \omega = 2b \)
4. \( \omega = 2\omega_0 \)

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

1. Two dimensional rigid body motion
2. Linear motion
3. Three dimensional rigid body motion
4. One dimensional rigid body motion

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.

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

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.

Explain overdamped, underdamped and critically damped harmonic motion.

Write an equation for a forced harmonic oscillator and obtain the conditions of resonance.

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.

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

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.