Rotational dynamics A disk rotating in a vertical plane abou

Rotational dynamics

A disk rotating in a vertical plane about a horizontal axis through its center has a cord wrapped around its outer edge and connected a mass m1= 0.200 kg. The disk has a mass M= 0.800 kg and a radius r= 0.150 m. The system is released from rest and the hanging mass falls a distance of 0.600 m in 1.50 seconds.

(a) Calculate the acceleration of the hanging mass.

(b) Calculate the angular acceleration of the disk

(c) Calculate the total acceleration of a point on the outer edge of the disk at the instant the hanging mass reaches a velocity of 0.500 m/s. Give the direction of this acceleration.

answers: (a) 3.27 m/s^2 (b) 21.8 rad/s^2 (c) 3.67 m/s^2

angular acc. angular acc. anqular vel.

Solution

Here ,

m1 = 0.200 Kg

M = 0.8 Kg

r = 0.150 m

distance ,d = 0.60 m

time , t = 1.50 s

a) let the acceleration of the mass is a

using second law of motion

a = m1 * g/(m1 + 0.5 * M * R^2/R^2)

a = 0.2 * 9.8/(0.2 + 0.5 * 0.8 * R^2/R^2)

a = 3.27 m/s^2

the acceleration of hanging mass is 3.27 m/s^2

b)

angular acceleration = a/R

angular acceleration = 3.27/(0.150)

angular acceleration = 21.8 rad/s^2

c)

at v = 0.5 m/s

at = v^2/r

at = 0.5^2/0.150

at = 1.67 m/s^2

total acceleartion = sqrt(1.67^2 + 3.27^2)

total acceleartion = 3.67 m/s^2

direction of acceleration with radius = arctan(3.27/1.67)

direction of acceleration with radius = 62.9 degree