WBJEE Physics - Mechanics and Thermal Physics Syllabus

Define relative velocity and give the formula for the velocity of A relative to B.

Relative velocity is the velocity of one body as observed from another. The velocity of A relative to B is $\vec{v}_{AB} = \vec{v}_{A} - \vec{v}_{B}$.

State the parallelogram law for the magnitude of the resultant of two vectors.

For vectors of magnitudes $A$ and $B$ at angle $\theta$: $$R = \sqrt{A^{2} + B^{2} + 2AB\cos\theta}$$ with direction $\tan\alpha = \frac{B\sin\theta}{A + B\cos\theta}$.

Define the scalar (dot) product and vector (cross) product of two vectors.

Dot product: $\vec{A}\cdot\vec{B} = AB\cos\theta$ (a scalar). Cross product: $\vec{A}\times\vec{B} = AB\sin\theta\,\hat{n}$ (a vector perpendicular to both, direction by right-hand rule).

State Newton's first law of motion and the concept it defines.

A body remains at rest or in uniform motion in a straight line unless acted upon by a net external force. It defines inertia and the concept of force as that which changes a body's state of motion.

State Newton's second law of motion in terms of momentum.

The rate of change of momentum is proportional to the applied force and acts in its direction: $$\vec{F} = \frac{d\vec{p}}{dt} = m\vec{a}$$ (for constant mass).

State Newton's third law of motion.

To every action there is an equal and opposite reaction; the forces of two bodies on each other are always equal in magnitude and opposite in direction, acting on different bodies.

Define linear momentum and state the law of conservation of momentum.

Linear momentum $\vec{p} = m\vec{v}$. If no net external force acts on a system, its total linear momentum remains constant: $\sum \vec{p}_{initial} = \sum \vec{p}_{final}$.

Define impulse and state the impulse-momentum theorem.

Impulse $\vec{J} = \vec{F}\,\Delta t$ (for constant force) or $\int \vec{F}\,dt$. The impulse-momentum theorem states $\vec{J} = \Delta \vec{p}$, i.e. impulse equals the change in momentum.

State the laws of limiting (static) and kinetic friction.

Limiting static friction: $f_{s} \leq \mu_{s} N$. Kinetic friction: $f_{k} = \mu_{k} N$. Friction is independent of contact area and opposes relative motion; generally $\mu_{s} > \mu_{k}$.

What is the relationship between the angle of friction $\phi$ and the coefficient of friction $\mu$?

$$\mu = \tan\phi$$ where $\phi$ is the angle between the resultant of normal reaction and limiting friction with the normal. The angle of repose equals the angle of friction.

Define angle of repose.

The angle of repose is the minimum angle of an inclined plane at which a body placed on it just begins to slide down. It equals the angle of friction: $\theta_{repose} = \tan^{-1}\mu_s$.

Give the formulas for centripetal acceleration and centripetal force in uniform circular motion.

Centripetal acceleration: $a_c = \frac{v^{2}}{r} = \omega^{2} r$. Centripetal force: $F_c = \frac{mv^{2}}{r} = m\omega^{2} r$, directed toward the centre.

For a car on a banked road (no friction), what is the relation for the ideal banking angle?

$$\tan\theta = \frac{v^{2}}{rg}$$ giving the optimum speed $v = \sqrt{rg\tan\theta}$ for which no friction is needed.

For a vehicle on a flat curved road, what is the maximum safe speed?

$$v_{max} = \sqrt{\mu r g}$$ where the friction force provides the necessary centripetal force.

Define work done by a constant force and state when it is zero.

$W = \vec{F}\cdot\vec{d} = Fd\cos\theta$. Work is zero when the force is perpendicular to the displacement ($\theta = 90^{\circ}$), or when displacement is zero, or force is zero.

State the work-energy theorem.

The net work done by all forces on a body equals the change in its kinetic energy: $$W_{net} = \Delta KE = \frac{1}{2}mv^{2} - \frac{1}{2}mu^{2}$$

Give the formulas for kinetic energy and its relation to momentum.

Kinetic energy $KE = \frac{1}{2}mv^{2}$. In terms of momentum $p$: $$KE = \frac{p^{2}}{2m}, \qquad p = \sqrt{2m\,KE}$$

Define average power and instantaneous power.

Average power $P_{avg} = \frac{W}{t}$. Instantaneous power $P = \frac{dW}{dt} = \vec{F}\cdot\vec{v}$. SI unit: watt (W); $1\,\text{hp} = 746\,\text{W}$.

Define a conservative force and give two examples.

A conservative force is one for which the work done depends only on the initial and final positions (independent of path) and the work in a closed loop is zero. Examples: gravitational force and spring (elastic) force.

What is the relation between a conservative force and its potential energy?

$$F = -\frac{dU}{dx}$$ The conservative force equals the negative gradient (slope) of the potential energy function.

Give the potential energy stored in a stretched spring and the gravitational PE near Earth's surface.

Spring PE: $U = \frac{1}{2}kx^{2}$ ($k$ = spring constant, $x$ = extension). Gravitational PE: $U = mgh$ (near Earth's surface).

State the law of conservation of mechanical energy.

In the absence of non-conservative forces (like friction), the total mechanical energy (kinetic + potential) of a system remains constant: $$KE + PE = \text{constant}$$

Distinguish between elastic and inelastic collisions.

In an elastic collision both momentum and kinetic energy are conserved. In an inelastic collision momentum is conserved but kinetic energy is not; in a perfectly inelastic collision the bodies stick together.

Define the coefficient of restitution $e$ and its values for elastic and perfectly inelastic collisions.

$$e = \frac{\text{relative velocity of separation}}{\text{relative velocity of approach}} = \frac{v_2 - v_1}{u_1 - u_2}$$ For a perfectly elastic collision $e = 1$; for a perfectly inelastic collision $e = 0$.

In a one-dimensional elastic collision, what happens to velocities when the two masses are equal?

When $m_1 = m_2$, the two bodies exchange velocities: the first takes the velocity of the second and vice versa.

Write the final velocities in a 1-D elastic collision of masses $m_1$ (velocity $u_1$) and $m_2$ (velocity $u_2$).

$$v_1 = \frac{(m_1 - m_2)u_1 + 2m_2 u_2}{m_1 + m_2}, \quad v_2 = \frac{(m_2 - m_1)u_2 + 2m_1 u_1}{m_1 + m_2}$$

Define the centre of mass of a system of particles.

The centre of mass is the point where the entire mass of the system may be assumed to be concentrated for describing its translational motion. Its position is $$\vec{R}_{cm} = \frac{\sum m_i \vec{r}_i}{\sum m_i}$$

Give the coordinates of the centre of mass of a two-particle system on the x-axis.

$$x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}$$ The centre of mass lies closer to the heavier particle.

How does the centre of mass of a system move under internal and external forces?

Internal forces do not affect the centre of mass. The centre of mass moves as if all the mass and total external force were concentrated at it: $$\vec{F}_{ext} = M \vec{a}_{cm}$$ If $\vec{F}_{ext} = 0$, the velocity of the centre of mass is constant.

For a projectile that explodes in mid-air, what path does its centre of mass follow?

Since the explosion involves only internal forces and gravity remains the only external force, the centre of mass continues along the original parabolic trajectory of the projectile.