Question

A green hoop with mass mh = 2.4 kg and radius Rh = 0.14 m hangs from a string that goes over a blue solid disk pulley with mass md = 2.3 kg and radius Rd = 0.08 m. The other end of the string is attached to an orange block on a flat horizontal surface that slides without friction and has mass m = 3.6 kg. The system is released from rest.
(a) What is magnitude of the linear acceleration of the hoop?
(b) What is magnitude of the linear acceleration of the block?
(c) What is the magnitude of the angular acceleration of the disk pulley?
(d) What are the tensions in the string between the block and disk pulley and between the hoop and disk pulley?
(e) What is the speed of the green hoop after it falls distance d = 1.49 m from rest.
(f) Now use work-energy conservation to solve for the hoop’s speed in (e).
(g) Now instead of the block, the other end of the string is attached to a massless axel through the center of an orange sphere that rolls without slipping and has mass m = 3.6 kg and radius R = 0.22 m (Figure 2). Use energy conservation principle to solve for the speed of the hoop after it falls distance d = 1.49 m from rest.

I'm completely lost. I know that the weight of the green hoop is the only force acting on our system. Therefore: $F_{net} = m_{h}g = 23.544 Newtons$
For part one, I know $\alpha_{n}=\frac{a}{R_{n}}$. However, I'm not sure how to find $\alpha_{n}$.

Any help is appreciated.

Answer 1

Answer:

a). linear acceleration of the hoop and block =  3.2895 m

                                                                                            s²

c). magnitude of the angular acceleration of the disk pulley = 41.119 rad

                                                                                                                  s²

d). tensions in the string between the block and disk pulley = 11.842 N

     tensions in the string between  the hoop and disk pulley = 15.625 N

check the pictures below for further explanation and for the remaining answers. I hope it helps you. Thank you

Explanation:

Start by writing "F=ma" equations for each of the things that moves. Also, since some of the objects (the pulley and the orange sphere) rotate, you should write "τ = Iα" equations (net torque = moment of inertia × angular acceleration) for those. In the end, you should have enough equations that you can combine them and solve for the desired quantities.

First, the hoop. There's no indication that it rotates, so we don't need a "τ = Iα" equation for it; just do "F=ma". The hoop has gravity ((mhoop)g) pulling down, and the tension in the vertical string (call it "T_v") pulling up.

Fnet = ma