R = 1.0 m
Angular acceleration = α = 5.0 rad/sec²
Instantaneous Angular velocity = ω(t) = α t = 5 t rad/sec
Instantaneous Velocity v(t) of the point at 1 m from the center of grinding wheel
= r ω(t) = 1 * 5 t
v (t) = 5 t m/sec
Instantaneous centripetal acceleration = a_r = v² / r = 25 t² / 1 = 25 t²
this is directed towards the center along the radius
Instantaneous linear (tangential) acceleration = r α
=> a_t = 1 m * 5 rad/sec² = 5 m/sec²
Instantaneous resultant acceleration = a = √[ a_r² + a_t² ]
a = √ [ 25 t² + 5² ]
a = 30.41 m/sec² at 6.0 sec
the direction of the angular velocity : tan⁻¹ [ a_t / a_r ]
= 1.59 deg. with the radius.
angular momentum = L = m v r = I ω
ω = m v r / I , where I is the moment of inertia.
Moment of inertia is increased by spreading the hands out...spreading the body out more. So the radius of gyration increases.
When the moment of inertia decreases, the angular velocity ω increases. When the skater curls himself/herself inside and sits, then his/her radius of gyration decreases. So moment of inertia decreases. So the skater spins very fast.