Mechanics

Definition: A second's pendulum is a simple pendulum with a period of exactly 2 seconds for one complete oscillation, so one swing (to and fro) takes 1 second.
Derivation: Period of simple pendulum: , where is length, is gravitational acceleration.
For T = 2 s:
Period of SHM: Maximum velocity cm/s, maximum acceleration cm/s².
Answer: Second's pendulum has T = 2 s; ; period = s.

When a spherical object falls through a viscous medium, three forces act:
1. Gravitational force: , where is the density of the object, is radius, is gravitational acceleration.
2. Buoyant force: , where is the density of the medium.
3. Viscous drag (Stokes’ law): , where is the viscosity, is velocity.
At terminal velocity , net force is zero:
Simplify:
Answer: .

Volume conservation: 8 droplets of radius form one drop of radius .
Surface area of one droplet: .
Total initial surface area (8 droplets): Surface area of final drop: Decrease in surface area: Answer: .

The moment of inertia ( ) of a rotating rigid body is defined as the mass property that quantifies its resistance to angular acceleration about a given axis. It is the sum of the products of the mass of each particle in the body and the square of its distance from the axis of rotation: , where is the mass of the -th particle and is its perpendicular distance from the axis.
SI Unit: kilogram meter squared ( ).
Dimensions: Using for mass, for length: Answer: Moment of inertia is the mass property resisting angular acceleration, . SI unit: , dimensions: .

For a uniform disc, the moment of inertia about an axis perpendicular to its plane through the center is: Given: , .
Answer: 1.8 kg m .

The critical velocity for fluid flow in a tube, where the flow transitions from laminar to turbulent, is related to Reynold's number . The formula is: where is the coefficient of viscosity, is the fluid density, and is the diameter of the tube.
Answer: .

Angular simple harmonic motion (SHM) occurs when the restoring torque is proportional to the angular displacement . The torque is , and using Newton’s second law for rotation, , where is the moment of inertia. Thus: where is the angular frequency.
Answer: .

At the uppermost position (top), tension and weight provide centripetal force: At the horizontal position, tension provides centripetal force: Assuming constant speed (minimum speed at top), let’s find minimum speed at top: (just enough to keep string taut).
At bottom (not horizontal, assuming typo for bottom as common in such problems):
Using energy conservation from top to bottom:
Difference: .
Answer: 6mg.

Capillary rise formula: , where is surface tension, is contact angle, is density, is gravity, and is radius.
Height is inversely proportional to radius: .
If radius is halved ( ):
Answer: 5 cm.