Derive an expression for kinetic energy of a rotating body with uniform angular velocity.
Question: Derive an expression for kinetic energy of a rotating body with uniform angular velocity.
To derive the expression for the kinetic energy of a rotating body with uniform angular velocity, we need to start with the formula for kinetic energy which is given by:
KE = (1/2)mv^2
where KE is the kinetic energy, m is the mass of the body, and v is its velocity.
For a rotating body, the velocity can be expressed in terms of its angular velocity, ω, and the radius, r, of the circle it is rotating in. The tangential velocity of a point on the rotating body is given by:
v = ωr
Substituting this expression for v into the definition of kinetic energy, we get:
KE = (1/2) m(ωr)^2
Simplifying, we get:
KE = (1/2) mω^2 r^2
This expression shows that the kinetic energy of a rotating body is proportional to the square of the angular velocity and the square of the radius of the circle the body is rotating in. This formula is useful in calculating the kinetic energy of rotating objects such as wheels, gears, and turbines in mechanical systems.
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