An green hoop with mass mh = 2.8 kg and radius Rh = 0.17 m hangs from a string t

Question

An green hoop with mass mh = 2.8 kg and radius Rh = 0.17 m hangs from a string that goes over a blue solid disk pulley with mass md = 2.4 kg and radius Rd = 0.08 m. The other end of the string is attached to a massless axel through the center of an orange sphere on a flat horizontal surface that rolls without slipping and has mass ms = 3.3 kg and radius Rs = 0.19 m. The system is released from rest. 1) What is magnitude of the linear acceleration of the hoop? m/s2 Your submissions: 0.383 Computed value:0.383Submitted:Friday, May 13 at 10:51 PM Feedback: 2) What is magnitude of the linear acceleration of the sphere? m/s2 3) What is the magnitude of the angular acceleration of the disk pulley? rad/s2 Your submissions: -47.6 Computed value:-47.6Submitted:Friday, May 13 at 10:47 PM Feedback: 4) What is the magnitude of the angular acceleration of the sphere? rad/s2 5) What is the tension in the string between the sphere and disk pulley? N 6) What is the tension in the string between the hoop and disk pulley? N 7) The green hoop falls a distance d = 1.56 m. (After being released from rest.) How much time does the hoop take to fall 1.56 m? s 8) What is the magnitude of the velocity of the green hoop after it has dropped 1.56 m? m/s 9) What is the magnitude of the final angular speed of the orange sphere (after the green hoop has fallen the 1.56 m)?

Explanation / Answer

Weight of the hoop = Fnet = 2.8kg*9.81m/s^2
Fnet = 27.468 N

The mass equivalent of M the pulley is found by
torque = F*R = I* = I*a/R
F = I*a/R^2
M*a = I*a/R^2
M = I/R^2
M = (1/2*m*R^2)/R^2
M = 1/2*m = 1/2*2.4
M = 1.2 kg

The mass equivalent of the rolling sphere is found by:
Moment of inertia of the sphere, I = 2/5*mR^2 for rotation about the center of mass + mR^2 for the distance of the axis of rotation from the center of mass of the sphere.
I = 7/5*mR^2
M = 7/5*m
M = 7/5 * 3.3
M = 4.62 Kg

the acceleration is then
a = F/m = 27.468/(2.8 + 1.2 + 4.62)
a = 3.19 m/s^2

1)
linear acceleration of hoop = 3.19 m/s^2

2)
linear acceleration of sphere = 3.19 m/s^2

3)
Angular acceleration disk pulley:
= a/R = 3.19/0.08
= 39.875 rad/s^2

4)
Angular acceleration sphere, = a/R
= 3.19/0.19
= 16.78 rad/s^2

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