JEE Physics Mechanics and Thermal Physics Notes Set 01

Motion, Gravitation and Rotation

1. Weight at height h: \(g' = g\left(\dfrac{R}{R+h}\right)^2\). Always use the distance from the centre of the earth.
2. Two-particle moment of inertia about the centre of mass: \(I = \mu d^2\) with \(\mu = \dfrac{m_1m_2}{m_1+m_2}\).
3. Rolling down an incline without slipping: \(a = \dfrac{g\sin\theta}{1 + I/mR^2}\). A body with smaller \(I/mR^2\) (solid sphere 2/5, disc 1/2, ring 1) arrives earlier, independent of mass and radius.
4. Centripetal acceleration \(a = \omega^2 r = v^2/r\). For equal periods it is proportional to the radius.
5. Tension in circular motion is perpendicular to velocity, so it does no work. Gravity does negative work while the bob rises and positive work while it falls.

Work, Momentum and Collisions

1. Momentum \(p = \sqrt{2mK}\). If K becomes 4 times, p doubles.
2. Force from a stream of bullets: \(F = n\,mv\), where n is the number fired per second.
3. If a ball keeps a fraction f of its energy after bouncing, the new height is f times the old height.

Properties of Matter and Waves

1. Strain under a force F: \(\dfrac{F}{AY}\), which varies as \(1/r^2\).
2. Terminal velocity \(v_T \propto r^2\). Merging n equal drops gives radius \(n^{1/3}r\).
3. Beats: \(|f_1 - f_2|\) per second. Loading a fork lowers its frequency, which settles the ambiguity between two possible values.
4. Simple pendulum \(T = 2\pi\sqrt{L/g}\): graph of L against T is a parabola, L against \(T^2\) a straight line.

Heat and Thermodynamics

1. Calorimetry: heat lost = heat gained, \(m_1c_1\Delta T_1 = m_2c_2\Delta T_2\).
2. Newton's law of cooling: rate of cooling is proportional to the excess of temperature over the surroundings.
3. Mixing gases at different temperatures: \(T = \dfrac{\sum n_iC_{v,i}T_i}{\sum n_iC_{v,i}}\). Use kelvin.
4. Adiabatic process \(PV^{\gamma} =\) constant. If \(P^a V^b\) is constant, then \(\gamma = b/a\).
5. Carnot engine: \(\eta = 1 - T_2/T_1\) and \(W = \eta Q_1\). Temperatures in kelvin.
6. Errors: for \(x = M^aL^bT^c\), the relative error is \(a\dfrac{\Delta M}{M} + b\dfrac{\Delta L}{L} + c\dfrac{\Delta T}{T}\) (magnitudes of the exponents).