Question

A cord is wound round the circumference of a wheel of radius 'r'. The axis of the wheel is horizontal and moment of inertia about it is T. A block of mass 'm' is attached to free end of the cord, initially at rest. When the wheel rotates and the block moves vertically downwards through distance 'h', the angular velocity of the wheel will be ______.
(Neglect the mass of cord, g =acceleration due to gravity)

पर्याय

• `((2"mgh")/("I" + "mr"^2))^(1//2)`

• `((2"gh")/("I" + "mr"))^(1//2)`

• (2 gh)^1/2

• `((2"mgh")/(1 + "2m"))^(1//2)`

Answer 1

A cord is wound round the circumference of a wheel of radius 'r'. The axis of the wheel is horizontal and moment of inertia about it is T. A block of mass 'm' is attached to free end of the cord, initially at rest. When the wheel rotates and the block moves vertically downwards through distance 'h', the angular velocity of the wheel will be `underline(((2"mgh")/("I" + "mr"^2))^(1//2))`.

Explanation:

Loss of potential energy of the mass 'm' is equal to the gain in kinetic energy of the wheel and the mass.

∴ `1/2 "I"omega^2 + 1/2 "mv"^2 = "mgh"`

∴ `1/2 "I"omega^2 + 1/2 "me"^2 omega^2 = "mgh"`

∴ `omega^2 = (2 "mgh")/("I" + "mr"^2)`