JEE Physics System Of Particles and Rotational Motion Notes Set 10

Angular Momentum

1. Angular momentum of a particle about a point: \( \vec L=\vec r\times\vec p \), of magnitude \( mvr\sin\theta=mv\times(\text{perpendicular distance of the line of motion})\). A particle moving in a straight line at constant velocity has constant angular momentum about every point.
2. A rigid body rotating about a fixed axis has \( L=I\omega \) about that axis.
3. The torque equation is \( \vec\tau_{ext}=d\vec L/dt \), valid about a fixed point or about the centre of mass.
4. Angular momentum about any point of a system equals the angular momentum of its centre of mass about that point plus the angular momentum about the centre of mass.

Conservation of Angular Momentum

If the net external torque about a point or an axis is zero, the angular momentum about it is conserved. Internal forces cannot change it. Common uses are the following.

1. A skater or a boy on a rotating stool spreads the arms: I increases, \( \omega \) falls, \( L=I\omega \) stays fixed (but the kinetic energy changes).
2. A body dropped gently on a rotating one (a ring on a disc): \( I_1\omega_0=(I_1+I_2)\omega \); kinetic energy is lost to friction.
3. Collisions with a hinged body: linear momentum is not conserved because the hinge gives an impulse, but angular momentum about the hinge is.
4. In a rough-surface problem the friction passes through the point of contact, so angular momentum about the contact point is conserved while the body slips.

Combined Translation and Rotation

For a body whose centre of mass moves with velocity \( v_{cm} \) while it spins with \( \omega \), the velocity of any point is \( \vec v=\vec v_{cm}+\vec\omega\times\vec r \) and the kinetic energy is \( K=\tfrac12Mv_{cm}^2+\tfrac12I_{cm}\omega^2 \). Newton's laws apply as \( \vec F=M\vec a_{cm} \) and \( \tau_{cm}=I_{cm}\alpha \), even if the centre of mass is accelerating (when torque is taken about the centre of mass).

Rolling Without Slipping

1. The contact point is at rest relative to the surface: \( v_{cm}=\omega R \) and \( a_{cm}=\alpha R \).
2. The body is turning about the contact point at that instant. The top point has speed \( 2v_{cm} \), the point level with the centre \( \sqrt2v_{cm} \); a point at angle \( \phi \) from the contact point has \( 2v_{cm}\sin(\phi/2) \).
3. Kinetic energy \( =\tfrac12Mv^2\left(1+\dfrac{k^2}{R^2}\right) \), where \( k^2/R^2 \) is 1 for a ring, 1/2 for a disc, 2/5 for a solid sphere, 2/3 for a hollow sphere.
4. Down an incline of angle \( \theta \): \( a=\dfrac{g\sin\theta}{1+k^2/R^2} \) and the friction needed is \( f=\dfrac{Mg\sin\theta\,(k^2/R^2)}{1+k^2/R^2} \), so rolling needs \( \mu\ge\dfrac{\tan\theta\,(k^2/R^2)}{1+k^2/R^2} \) (for a solid sphere \( \tfrac27\tan\theta \)).
5. Static friction does no work in pure rolling, so mechanical energy is conserved. The body with the smallest \( k^2/R^2 \) reaches the bottom first: solid sphere, disc, hollow sphere, ring in that order.

Slipping Changing to Rolling

If a body is placed on a rough surface with \( v_{cm}\ne\omega R \), kinetic friction \( \mu Mg \) acts until \( v_{cm}=\omega R \). It reduces the linear speed and changes the spin until pure rolling begins. Use angular momentum about the contact point to find the final speed: for example, a ring thrown with speed \( V_0 \) and backspin \( V_0/2R \) ends rolling with \( V_0/4 \); a body given spin but no velocity always moves in the sense in which the bottom point is being pushed.

Instantaneous Axis of Rotation

At any instant the motion of a rigid body in a plane is a pure rotation about one point, the instantaneous centre. It is found where the perpendiculars to the velocities of two points meet. For a ladder sliding on a wall and a floor it is the point of intersection of the vertical through the top end and the horizontal through the foot.

Toppling

1. A body resting on a rough surface and pushed by a horizontal force F tilts about the lower edge when the torque of F exceeds the torque of the weight about that edge. For a block of base width b pushed at height h: \( F_{topple}=\dfrac{mgb}{2h} \).
2. The normal force shifts towards the edge as F grows; toppling begins when it reaches the edge.
3. The body slides before it topples if the limiting friction \( \mu mg \) is reached before F reaches \( F_{topple} \): for a block of base width b pushed at height h, sliding comes first when \( \mu<\dfrac{b}{2h} \).
4. On an incline the block slides at \( an heta=\mu \) and topples at \( an heta=\dfrac{ ext{base}}{ ext{height}} \).

Worked Examples

Example 1. A ring of mass 2 kg and radius 0.5 m rolls without slipping with speed 3 m/s. Find its total kinetic energy.

1. For a ring \( K=\tfrac12mv^2(1+1)=mv^2 \).
2. \( K=2\times9=18 \) J.

Example 2. A solid sphere rolls down a 30° incline without slipping. Find the acceleration and the minimum friction coefficient (g = 10 m/s²).

1. \( a=\dfrac{g\sin\theta}{1+2/5}=\dfrac{5}{1.4}=3.57 \) m/s².
2. \( \mu_{min}=\tfrac27\tan30^\circ=0.165 \).

Example 3. A disc of moment of inertia 2 kg m² rotates at 10 rad/s. A ring of moment of inertia 3 kg m² at rest is gently dropped on it coaxially. Find the final angular speed and the energy lost.

1. \( 2\times10=(2+3)\omega\Rightarrow\omega=4 \) rad/s.
2. Energy lost \( =\tfrac12(2)(100)-\tfrac12(5)(16)=100-40=60 \) J.

Example 4. A uniform cube of side 0.4 m and mass 10 kg is pushed by a horizontal force at height 0.3 m. Find the least force that topples it (friction large).

1. Torque about the lower edge: \( F\times0.3=mg\times0.2 \).
2. \( F=\dfrac{10\times10\times0.2}{0.3}=66.7 \) N.

Example 5. A uniform ring of radius R is thrown with speed \( V_0 \) and backspin \( V_0/2R \). Find the speed when pure rolling starts.

1. Angular momentum about the contact point: \( mV_0R-mR^2\dfrac{V_0}{2R}=\dfrac12mV_0R \).
2. Finally \( mVR+mR^2\dfrac VR=2mVR \Rightarrow V=\dfrac{V_0}{4} \).

Quick Revision Points

1. \( \tau_{ext}=dL/dt \); no torque means \( L \) is conserved.
2. Rolling: \( v=\omega R \), kinetic energy \( \tfrac12Mv^2(1+k^2/R^2) \).
3. Acceleration on an incline \( g\sin\theta/(1+k^2/R^2) \).
4. Slipping changing to rolling: conserve angular momentum about the contact point.
5. Toppling: torque of the push about the edge against torque of the weight.