An green hoop with mass m h = 2.7 kg and radius R h = 0.15 m hangs from a string

Question

An green hoop with mass mh = 2.7 kg and radius Rh = 0.15 m hangs from a string that goes over a blue solid disk pulley with mass md = 2.2 kg and radius Rd = 0.09 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 = 3.8 kg and radius Rs= 0.21 m. The system is released from rest.

1)

What is magnitude of the linear acceleration of the hoop?

2)

What is magnitude of the linear acceleration of the sphere?

3)

What is the magnitude of the angular acceleration of the disk pulley?

4)

What is the magnitude of the angular acceleration of the sphere?

5)

What is the tension in the string between the sphere and disk pulley?

6)

What is the tension in the string between the hoop and disk pulley?

7)

The green hoop falls a distance d = 1.59 m. (After being released from rest.)

How much time does the hoop take to fall 1.59 m?

8)

What is the magnitude of the velocity of the green hoop after it has dropped 1.59 m?

9)

What is the magnitude of the final angular speed of the orange sphere (after the green hoop has fallen the 1.59 m)?

Explanation / Answer

Mh*(G-a) - T = (1/2)*Md*a
The T in the above equation is the tension between the sphere and the disk.
T = m*a + I*alpha/R
= (Ms*a) + (2/5)(Ms*Rs^2)*(a/Rs)/Rs
= (Ms*a) + (2/5)(Ms*a)
T = (7/5)(Ms*a)

Using the two equations, solve for a:
Mh*(G-a) - (7/5)(Ms*a) = (1/2)*Md*a
Mh*G = ((1/2)*Md*a + (7/5)(Ms*a) + Mh*a) = (0.5Md+1.4Ms+Mh)a
a = Mh*G/(0.5Md+1.4Ms+Mh)

a=2.7*9.8/(0.5*2.2+1.4*3.8+2.7)=26.46/8.13=3.254612

2. I'm assuming linear acceleration is the same for the whole system, so then it should be the same answer for this question.

3.It should be the linear acceleration divided by the radius of the disk/pulley due to the equation

a = alpha*R.

angular acceleration=3.254612/0.09=36.16236162

4.angular acceleration=3.254612/0.21=15.498154981

5.T = (7/5)(Ms*a)=17.31453584

7.s=u*t+0.5*a*t*t

as u=0

calculate t

8.v^2-u^2=2as

u=0

calculate v

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