JEE Physics Laws Of Motion Notes Set 03

Laws of Motion and Friction: Quick Revision

Newton's first law says a body stays at rest or in uniform straight-line motion unless a net external force acts on it; mass measures this inertia. The second law is \( \vec F=\frac{d\vec p}{dt} \), which for constant mass becomes \( \vec F=m\vec a \). The third law says forces occur in equal and opposite pairs acting on two different bodies, so they never cancel each other in the free body diagram of one body. Impulse \( \int F\,dt \) equals the change in momentum, and it is the area under a force-time graph.

Method for Connected Bodies

1. Draw a free body diagram of each body and choose a positive direction along the motion.
2. For bodies in contact or joined by a taut string, the accelerations are equal (or related by the geometry of the pulley).
3. The tension or contact force at a joint is the mass beyond that joint times the common acceleration (for a row of blocks on a smooth surface).
4. Two masses over a smooth fixed pulley: \( a=\frac{(m_1-m_2)g}{m_1+m_2} \) and \( T=\frac{2m_1m_2g}{m_1+m_2} \). For a mass on a smooth table pulled by a hanging mass: \( a=\frac{m_2g}{m_1+m_2} \) and \( T=\frac{m_1m_2g}{m_1+m_2} \).
5. A massless string has the same tension throughout; a rope with mass has tension that changes along its length.

Pulleys, Lifts and Pseudo Force

If a pulley is raised with acceleration a, replace g by \( g+a \) in the formulas. In a frame accelerating with \( \vec a_0 \) a pseudo force \( -m\vec a_0 \) is added. A man in a lift of acceleration a upward reads an apparent weight \( m(g+a) \); downward acceleration a gives \( m(g-a) \), which is zero in free fall. A spring balance on a pulley with masses \( m_1,m_2 \) reads \( \frac{4m_1m_2}{m_1+m_2} \) (in force units of g), which is less than \( m_1+m_2 \) while they accelerate. A spring force cannot change instantly, but a taut string tension can, which matters when a string is cut.

Friction

Static friction adjusts itself to the applied force up to a limiting value \( f_s\le\mu_sN \). Kinetic friction is \( f_k=\mu_kN \), usually a little less. On a rough horizontal surface, pulling at angle \( \theta \) above the horizontal gives \( N=mg-F\sin\theta \) and pushing gives \( N=mg+F\sin\theta \), so pulling is easier. The minimum force to start motion is \( \frac{\mu mg}{\sqrt{1+\mu^2}} \) at \( \tan\theta=\mu \). The angle of repose satisfies \( \tan\theta_R=\mu_s \). A block on an incline moves down with acceleration \( g(\sin\theta-\mu\cos\theta) \). If the block rests on the wedge face with no friction, the wedge needs horizontal acceleration \( g\tan\theta \) to keep the block stationary relative to it.

Two-Block Friction Problems

For a block resting on another block on a smooth floor, the upper block is accelerated only by friction, so the common acceleration cannot exceed \( \mu g \). The maximum push is then \( (m_1+m_2)\mu g \). Check each friction limit against the force that tries to move that block before assuming they slide together or separately.