JEE Physics Laws Of Motion Notes Set 04

Newton's Laws and Free-Body Diagrams

The first law defines inertia: a body keeps its state of rest or uniform motion until a net external force acts on it. The second law gives \( \vec F_{net}=\frac{d\vec p}{dt} \), which reduces to \( \vec F=m\vec a \) for constant mass. The third law says forces occur in equal and opposite pairs that act on different bodies, so an action-reaction pair can never cancel on a single body.
For every problem, isolate each body, draw all forces acting on it, choose axes along the acceleration, and write \( \sum F=ma \) for each body. Weight and normal reaction on a block at rest are not an action-reaction pair; they are two forces on the same body that happen to balance.

Strings, Pulleys and Constraint Relations

An ideal string is massless and inextensible, so it carries the same tension throughout and connected points have equal speed along the string. For an ideal pulley the tension is the same on both sides. For two masses over a fixed pulley, \( a=\frac{(m_1-m_2)g}{m_1+m_2} \) and \( T=\frac{2m_1m_2g}{m_1+m_2} \).
A movable massless pulley supported by a force F has string tension F/2 on each side. Constraint rule: the total length of string is fixed, so the sum of the displacements of the string segments (taken with proper signs) is zero; differentiate twice to get the acceleration relation. A rope of mass m pulling a block of mass M with force P exerts \( \frac{PM}{M+m} \) on the block.
Springs cannot change their force instantly, while strings can: just after a string is cut the spring force is unchanged.

Friction

Static friction adjusts itself up to a limiting value \( f_s\le\mu_sN \). Once sliding begins, kinetic friction \( f_k=\mu_kN \) acts, and \( \mu_k<\mu_s \). The contact force is \( \sqrt{N^2+f^2} \), and the angle of friction satisfies \( \tan\lambda=\mu \).
For a block on an incline, it remains at rest while \( \tan\theta\le\mu_s \). The force along the plane needed to push it up at the verge of motion is \( mg(\sin\theta+\mu\cos\theta) \), and to hold it from sliding down is \( mg(\sin\theta-\mu\cos\theta) \). For a block on top of another, find the common acceleration first and check whether the required friction exceeds the limiting value; if it does, the blocks slip and kinetic friction acts. Pulling at a small angle upward lowers N and hence friction; pushing downward raises it.

Equilibrium, Impulse and Variable Mass

For translational equilibrium the net force is zero, and for rotational equilibrium the net torque is zero. Three concurrent forces in equilibrium obey Lami's theorem. Impulse is \( \int F\,dt \) and equals the change in momentum; for a force that varies with time, it is the area under the force-time graph.
For a stream of matter hitting a surface at rate \( \frac{dm}{dt} \) with relative speed \( v_r \), the force is \( v_r\frac{dm}{dt} \), and this becomes \( \rho A v_r^2 \) for a jet that is stopped. A body picking up mass at rest feels a retarding force \( v\frac{dM}{dt} \). In an accelerated frame, add a pseudo force \( -m\vec a_{frame} \) to each body; this turns problems on accelerated wedges, carts and lifts into equilibrium problems.