JEE Physics System Of Particles and Rotational Motion Notes Set 06

Centre of Mass

1. For particles: \( \vec r_{cm}=\dfrac{\sum m_i\vec r_i}{\sum m_i} \). For a continuous body: \( \vec r_{cm}=\dfrac{1}{M}\int\vec r\,dm \).
2. The centre of mass always lies within the body or the group of particles (inside their convex hull), though it may lie outside the material, as in a ring.
3. For a body with a hole, treat the removed part as a negative mass at its own centre. For a disc of radius R with a hole of radius a whose centre is at distance d, the centre of mass shifts by \( \dfrac{a^2d}{R^2-a^2} \) away from the hole.
4. Group particles first: replace any sub-group by its total mass at its own centre of mass.

Motion of the Centre of Mass

1. Total momentum is \( \vec P=M\vec v_{cm} \) and \( \vec F_{ext}=M\vec a_{cm} \).
2. Internal forces (springs, explosions, collisions between parts) cannot change the motion of the centre of mass.
3. If the external force is zero, the velocity of the centre of mass is constant. A body or system that starts at rest keeps its centre of mass at rest, so if one part moves one way the other moves the other way.
4. A projectile that explodes in flight: the centre of mass of the fragments continues on the original parabola until some fragment lands.
5. Man on a boat or trolley on a smooth surface: \( m_1x_1=m_2x_2 \) for the displacements relative to the ground, and the relative displacement is \( x_1+x_2 \).
6. Two-body relative motion (spring, string) can be handled with the reduced mass \( \mu=\dfrac{m_1m_2}{m_1+m_2} \).

Collisions

1. Definition: \( e=\dfrac{\text{relative speed of separation}}{\text{relative speed of approach}} \) along the line of impact.
2. Head-on elastic collision with the second body at rest: \( v_1=\dfrac{m_1-m_2}{m_1+m_2}u_1,\ v_2=\dfrac{2m_1}{m_1+m_2}u_1 \). Equal masses exchange velocities. A heavy body hitting a light body barely slows; a light body hitting a heavy body rebounds.
3. The fraction of kinetic energy transferred in the elastic case is \( \dfrac{4m_1m_2}{(m_1+m_2)^2} \), and it is greatest (equal to 1) for equal masses.
4. Perfectly inelastic collision: the bodies move together with \( v=\dfrac{m_1u_1+m_2u_2}{m_1+m_2} \).
5. Ball dropped from height h onto the ground: rebound speed \( eu \); height after the nth rebound \( he^{2n} \); \( e=\sqrt{h_1/h} \).
6. Oblique collision with a smooth surface: the component along the surface is unchanged and the normal component is multiplied by \( -e \). With e = 1 the angle of incidence equals the angle of reflection; for e less than 1 the ball leaves at an angle to the normal that is larger than the incoming angle.
7. Collision with a moving heavy surface: work in the frame of the surface, reverse the relative normal velocity (times e), then go back to the ground frame.

Ballistic Pendulum and Typical Set-ups

1. A bullet of mass m and speed v embeds in a block of mass M hanging by a string: \( mv=(m+M)V \), and the block rises to a height y with \( V=\sqrt{2gy} \), so \( v=\dfrac{m+M}{m}\sqrt{2gy} \).
2. Gun and bullet: the momenta are equal and opposite, so the lighter object moves faster. The recoil of the gun is \( mv/M \).
3. Ball sliding in a groove of a free block on a smooth floor: horizontal momentum is conserved and the energy is shared, so the ball is slower than it would be on a fixed block.
4. Explosion into fragments: the vector sum of the momenta is zero if the body was at rest, and the energy released is the total kinetic energy of the fragments.
5. Block hit while hanging from a spring: during the collision the spring force and gravity are unchanged and finite, so only the momentum equation is used, and the acceleration just afterwards comes from the net force.