Physics (Mechanics) - B.Tech 1st Semester Exam., 2021

The position of a particle of mass \( m \) under the influence of a free particle is given by \( \vec{r} = A \sin \omega t \hat{i} + B \cos \omega t \hat{j} \). Find the expression for its force.

Express \( \vec{s} \) of cylindrical coordinate system into unit vectors of Cartesian coordinate system.

Give two examples of conservative forces.

Define Euler angles.

Consider a cloud of point particles interacting through gravitational forces and having a distribution of kinetic energy. What is the condition of potential energy under which this cloud will expand?

Why is a Foucault pendulum situated at the equator will not detect the rotation of the earth about its axis?

A condenser of capacity \( 1 \, \mu F \), an inductance \( 0 \cdot 2 \, H \) and a resistance of \( 100 \, \Omega \) are in series. Is the circuit oscillator? Why?

A pendulum is of length 50 cm. Find its period when it is suspended in (i) a stationary lift and (ii) a lift falling at a constant velocity of 5 m/s.

Write down the expression for moment of inertia of a solid circular disk, axis perpendicular to its plane and passing through the centre.

Fill in the blank : Gradient of scalar field is ______ to the equipotential surfaces.

A particle moves with \( \dot{\theta} = \omega = \) constant and \( r = r_0 e^{\beta t} \), where \( r_0 \) and \( \beta \) are constants. For what values of \( \beta \), the radial part of acceleration vanishes?

A point is observed to have velocity \( V_A \) relative to coordinate system \( A \). What is its velocity relative to coordinate system \( B \), which is displaced from system \( A \) by a distance \( R \)? (\( R \) can change in time.)

A truck at rest has one door fully open, as shown below. The truck accelerates forward at constant rate \( A \), and the door begins to swing shut. The door is uniform and solid, has total mass \( M \), height \( h \), and width \( w \). Neglect air resistance. Find the instantaneous angular velocity of the door about its hinges when it has swung through \( 90^{\circ} \).

Using cylindrical coordinate system, find out the volume of a cylinder of radius \( R \) and height \( H \).

What are conservative and non-conservative forces? Show that the electrostatic force between two charges \( q_1 \) and \( q_2 \) placed at a distance of \( r \) are conservative. Also, obtain an expression for the potential energy of two charges.

State and discuss Kepler's law. Show that the Newton's law can be deduced from Kepler's law.

Discuss the energy equation in the centre of mass system.

Derive Euler's equations of rigid body motion and discuss their necessity in describing rigid body.

Explain, why flying saucers make better spacecraft than do flying cigars.

Discuss three-dimensional rigid body motion describing angular velocity and moment of inertia tensor.

A thin hoop of mass \( M \) and radius \( R \) rolls without slipping about the z-axis. It is supported by an axle of length \( R \) through its centre. The hoop circles around the z-axis with angular speed \( \Omega \). What is the instantaneous angular velocity of the hoop? Given, the moment of inertia of a hoop for an axis along its diameter is \( \frac{1}{2} MR^2 \).

Discuss three-dimensional rigid body motion describing angular velocity and moment of inertia tensor.

A thin hoop of mass M and radius R rolls without slipping about the z-axis. It is supported by an axle of length R through its centre, as shown below: The hoop circles around the z-axis with angular speed \(\omega \). What is the instantaneous angular velocity of the hoop? Given, the moment of inertia of a hoop for an axis along its diameter is\((1/2) MR^2 \).