xSkill: notes, lectures and practice for CBSE and CSE

3.1 Impulse A force that acts for a short time changes the momentum by an amount called the impulse. Impulse is the force multiplied by the time for a constant force, and the area under the force-time curve for a varying force. The impulse-momentum theorem states that the impulse on a body equals its change in momentum, and it is just the second law integrated over time. The theorem explains the design of safety devices. A cricket player draws the hands back while catching, so the same change in momentum happens over a longer time and the force on the hands is smaller. Airbags, crumple zones and landing mats all do the same job: they stretch the time over which the momentum change occurs. A hard stop delivers a large force, a soft stop delivers a small one, and the impulse is the same in both cases.

J = F_{a v} Δ t = Δ p, F_{a v} = \frac{Δ p}{Δ t}

Impulse and the average force over the contact time

3.2 Conservation of momentum In a system with no net external force, the total momentum before an event equals the total momentum after it. The internal forces during a collision are equal and opposite pairs, so they change the momenta of the individual bodies but not the total. This gives a powerful method for collision and explosion problems where the forces are unknown but the total momentum is known. A gun and bullet is the standard example. Before firing, the total momentum is zero. After firing, the bullet carries momentum forward and the gun carries equal momentum backward, so the recoil speed of the gun is the bullet's momentum divided by the gun's mass. A rocket works on the same principle: the exhaust gases carry momentum backward and the rocket moves forward, even in empty space. Momentum is conserved in every direction separately. In a two dimensional collision, write the conservation equation along x and along y, and remember that kinetic energy need not be conserved even when momentum is. A completely inelastic collision loses the maximum kinetic energy while still obeying momentum conservation, and an elastic collision conserves both. The deciding quantity is the coefficient of restitution, which measures how much of the relative approach speed is recovered after the impact. A value of one means a perfectly elastic collision and a value of zero means the bodies stick together after the impact, so the coefficient shows how kinetic energy is shared after the event.

m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}

Conservation of total momentum in a collision, in vector form

3.3 Equilibrium of a particle A particle is in equilibrium when the net force on it is zero. In two dimensions this gives two scalar equations, one for each axis, so at most two unknown forces can be found from it. When three forces act at a point and the body stays at rest, the three vectors must form a closed triangle, and the sine rule applied to that triangle is known as Lami's theorem. The general method is the free body diagram. Isolate the body, draw every force acting on it with its direction, choose axes, and write the second law with zero acceleration for equilibrium. Keep the components of each force separate and never add forces as plain numbers. A sketch with the angles marked is worth as much as the algebra, because a wrong angle ruins the components even when the equations are written correctly. For three concurrent forces, checking that the vector sum closes the triangle is a quick check on the algebra. When the problem has more than two unknowns, look for a second body whose diagram supplies another equation, since one body alone gives only two equilibrium conditions in a plane.

\sum F_{x} = 0, \sum F_{y} = 0, \frac{F _{1}}{sin α _{1}} = \frac{F _{2}}{sin α _{2}} = \frac{F _{3}}{sin α _{3}}

Equilibrium equations in two dimensions and Lami's theorem for three forces

3.4 Worked example: a hanging sign A sign of weight 200 N hangs from two wires that make equal angles of 30 degrees with the ceiling. By symmetry the two tensions are equal. The vertical components must balance the weight, so 2T sin 30 equals 200, which gives T equal to 200 N. The horizontal components are T cos 30 each and they are equal and opposite, so they cancel as they must. If the wires were made more horizontal, the tension would rise sharply because the sine of a small angle is small, which is why a clothesline sags and a tightrope needs huge tension for a small weight. This sensitivity is worth remembering in estimation questions and in the analysis of a hanging lamp. A small increase in weight near a nearly horizontal string produces a large jump in tension, and a wire that looks strong enough can still fail. In a three force equilibrium, adding the vectors head to tail closes a triangle, and the law of sines converts that triangle into the force magnitudes. A body with four or more concurrent forces needs the component method instead of the triangle.

Warning

Equilibrium means zero acceleration, not zero velocity. A car moving at constant velocity on a straight road is in equilibrium too, so the net force on it is zero even though it is moving.