JEE Physics System Of Particles and Rotational Motion Notes Set 03

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Rotational Kinematics

For rotation about a fixed axis the angle \( \theta \), angular velocity \( \omega = d\theta/dt \) and angular acceleration \( \alpha = d\omega/dt \) play the role of x, v and a. For constant \( \alpha \): \( \omega = \omega_0 + \alpha t \), \( \theta = \omega_0t + \tfrac12\alpha t^2 \), \( \omega^2 = \omega_0^2 + 2\alpha\theta \). A point at distance r from the axis has \( v = \omega r \), tangential acceleration \( a_t = \alpha r \) and centripetal acceleration \( \omega^2r \).

Moment of Inertia

The moment of inertia \( I = \sum m_ir_i^2 \) measures the resistance to change in rotation and depends on the axis. Parallel axes theorem: \( I = I_{cm} + Md^2 \). Perpendicular axes theorem (planar bodies only): \( I_z = I_x + I_y \). The radius of gyration k is defined by \( I = Mk^2 \).

Body and axisI
Thin rod, centre, perpendicular\( ML^2/12 \)
Thin rod, end, perpendicular\( ML^2/3 \)
Ring, axis perpendicular through centre\( MR^2 \)
Disc, axis perpendicular through centre\( MR^2/2 \)
Solid sphere, diameter\( 2MR^2/5 \)
Hollow sphere, diameter\( 2MR^2/3 \)

Torque and Equilibrium

Torque \( \vec\tau = \vec r\times\vec F \) has magnitude \( rF\sin\theta \). For fixed-axis rotation \( \tau = I\alpha \). A body is in equilibrium when the net force and the net torque about any point are both zero. For a ladder or beam, pick the point that removes the most unknown forces and balance torques there. A couple is a pair of equal and opposite forces with torque F times the separation.

System of Particles and Rotational Motion Concepts and Summary for JEE Physics

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