Shortcut Methods
Shortcut Methods and Tricks for Solving Numerical Problems on Center of Mass - System of Particles and Rotational Motion:
JEE Main and Advanced:
- Center of Mass of a Uniform Rod with a Particle: Divide the problem into two parts: the rod and the particle. Find the center of mass of the rod as if the particle were not there, and then find the center of mass of the particle as if the rod were not there. Finally, use the formula for the center of mass of a system of two particles:
$$ \bar{x} = \frac{m_1x_1 + m_2x_2}{m_1 + m_2}$$ where (m_1) and (m_2) are the masses of the rod and the particle, respectively, and (x_1) and (x_2) are their respective coordinates (measured from the pivot).
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Velocities of Two Particles after String Cut: Use the law of conservation of momentum along the direction of motion. The total momentum of the system before the string is cut is zero (since the particles are at rest), so the total momentum after the string is cut must also be zero. This gives you an equation that you can solve for the velocities of the particles.
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Kinetic Energy of a Rotating Disc: Use the formula for the kinetic energy of a rotating object:
$$K = \frac{1}{2}I\omega^2$$ where (K) is the kinetic energy, (I) is the moment of inertia, and (\omega) is the angular velocity.
- Total Kinetic Energy of a Rolling Sphere: Use the formula for the total kinetic energy of a rolling object:
$$K = \frac{1}{2}(mv^2 + I\omega^2)$$ where (K) is the total kinetic energy, (m) is the mass of the sphere, (v) is its velocity, (I) is the moment of inertia, and (\omega) is the angular velocity.
- Angular Momentum of a Particle in Circular Motion: Use the formula for the angular momentum of a particle in circular motion:
$$L = mvr$$ where (L) is the angular momentum, (m) is the mass of the particle, (v) is its speed, and (r) is the radius of the circular path.
- Angular Momentum of a Rigid Body: Use the formula for the angular momentum of a rigid body rotating about a fixed axis:
$$L = I\omega$$ where (L) is the angular momentum, (I) is the moment of inertia, and (\omega) is the angular velocity.
CBSE Board Exams:
- Angular Acceleration of a Uniform Rod: Use the formula for the torque on a rigid body rotating about a fixed axis:
$$ \tau = I\alpha $$
where (\tau) is the torque, (I) is the moment of inertia, and (\alpha) is the angular acceleration.
- Acceleration of a Particle on a Rotating Disc: Use the formula for the centripetal acceleration of a particle in circular motion:
$$a_c = \frac{v^2}{r}$$ where (a_c) is the centripetal acceleration, (v) is the speed of the particle, and (r) is the radius of the circular path.
- Acceleration of a Rolling Sphere: Use the formula for the acceleration of a rolling object on an inclined plane:
$$a = g\sin\theta$$ where (a) is the acceleration, (g) is the acceleration due to gravity, and (\theta) is the angle of the inclined plane.
- Centripetal Force:
$$F_c = mv^2/r$$
where (F_c) is the centripetal force acting on the particle of mass (m) moving with speed (v) along a path with a radius (r).
- Angular Acceleration of a Rigid Body: Use the formula for the torque on a rigid body rotating about a fixed axis:
$$ \tau = I\alpha $$
where (\tau) is the torque, (I) is the moment of inertia, and (\alpha) is the angular acceleration.
