JEE Physics Laws Of Motion Notes Set 02

Force and Newton's Laws

1. A force is a push or pull that changes, or tries to change, the state of motion of a body. Net force is the vector sum of all forces acting on it.
2. First law: a body stays at rest or in uniform motion along a straight line unless a net external force acts on it. This idea defines inertia, and mass is the measure of inertia.
3. Second law: the net force equals the rate of change of momentum, \(\vec F=\frac{d\vec p}{dt}\). For constant mass, \(\vec F=m\vec a\).
4. Third law: when body A pushes on body B, B pushes back on A with an equal and opposite force. The two forces act on different bodies, so they never cancel each other.
5. Impulse is \(\vec J=\int\vec F\,dt=\Delta\vec p\); on a force-time graph it is the area under the curve, and areas below the axis count as negative.

Solving Problems with Free-Body Diagrams

1. Draw the body alone, mark every force on it (weight, normal reaction, tension, friction, applied force) and choose axes along and perpendicular to the expected motion.
2. Write \(\sum F=ma\) along each axis for each body. For bodies joined by a taut string the accelerations are related by a constraint.
3. An ideal (massless, inextensible) string has the same tension throughout; a smooth ideal pulley only changes the direction of the tension.
4. For a movable pulley, a body hung from it moves half as far as the string end, so its acceleration is half. Always write the length of the string as a constant and differentiate twice.
5. Bodies in contact: the force between two blocks must accelerate everything behind that contact. For three blocks of masses \(m_1,m_2,m_3\) pushed by F, the force between the last two is \(m_3\frac{F}{m_1+m_2+m_3}\).

Standard Results

Friction

1. Static friction adjusts itself to the applied force up to a maximum \(f_s\le\mu_sN\). Once the body slides, kinetic friction \(f_k=\mu_kN\) acts, and \(\mu_k\) is slightly smaller than \(\mu_s\).
2. Friction does not depend on the area of contact, and it opposes the relative (or impending relative) motion of the surfaces. Static friction is larger than sliding friction, which is larger than rolling friction.
3. Angle of friction \(\lambda\): \(\tan\lambda=\mu\). The resultant of N and friction makes this angle with the normal at the point of slipping.
4. On a rough incline: a body just begins to slide when \(\tan\theta=\mu_s\). Sliding down, \(a=g(\sin\theta-\mu_k\cos\theta)\); sliding up, the retardation is \(g(\sin\theta+\mu_k\cos\theta)\).
5. The total contact force from a rough surface is \(\sqrt{N^2+f^2}\), which lies between N and \(N\sqrt{1+\mu^2}\).
6. A block pressed against the vertical front of an accelerating cart stays up if \(\mu ma\ge mg\), that is \(a\ge\frac{g}{\mu}\).
7. Stopping distance of a vehicle braking with locked wheels: \(s=\frac{u^2}{2\mu g}\).

Frames of Reference

1. Newton's laws hold in inertial frames. In a frame with acceleration \(\vec a_0\), add a pseudo force \(-m\vec a_0\) to every body.
2. With the pseudo force included, the same free-body method works: a bead on a parabolic wire \(y=kx^2\) in a frame accelerating at a stays at rest where the slope \(\frac{dy}{dx}=\frac{a}{g}\), that is \(x=\frac{a}{2gk}\).
3. For constraints (rods sliding against a wall and floor), the velocity components along the rod are equal at both ends.