How Long Does It Take a Block to Move 30cm with a Frictionless Pulley System?

[onaboat]

A 1.5 kg block is connected by a rope across a 50 cm-diameter, 2.0 kg, frictionless pulley. A constant 10 N tension is applied to the other end of the rope. Starting from rest, how long does it take the block to move 30cm?

I summed up the force of the tension and the force of gravity of the block. I defined the direction the tension was going to be positive which would make the force of gravity be negative. Summing them up I got -4.7 N and then using Newton's 2nd law I found the acceleration to be -3.13 m/s^2. Knowing Vo=0, d=.3m, a=-3.13 m/s^s and using the kinematic equation: d=Vo*t+.5*a*t^2. Plugging in everything I know I got the time to be .44s. This wasn't the right answer and I have no idea what to do now. I'm thinking that the torque on the pulley might be involved but I can't tell how to use it.

 

[Vyse007]

Yup you got it right...the torque will be involved.
We know that the torque is the product of force and perpendicular distance. Hence, the torque on the pulley will be the tension applied x the radius of the pulley (in m). Draw a FBD again taking this into account. Let us know if you get stuck.

 

[onaboat]

I found the torque to be 2.5 N*m. Then using $$T$$=I$$\alpha$$ I found the angular acceleration to be 20 rad/s^2. Then using s=$$\Theta$$*r I found the distance the pulley moved was 1.2 radians. I then used the circular motion for of the kinematic equation I used before and got .35s. Does this seem right?

 

[Vyse007]

That does seem right...I am not talking about the calculations, but the method you employed.
Though I think that it could be solved by taking linear acceleration from the angular acceleration too.
Anyways, have you checked your answer?

 

[rwisz]

I would first double check the calculations and make sure all units are consistent. It is important to also consider the direction of motion and make sure the signs are correct in the equations. In this case, the tension should be positive and the force of gravity should be negative.

Additionally, it is important to note that the tension in the rope will change as the block moves, so the acceleration will also change. This means that the kinematic equation used may not accurately represent the motion of the block.

To solve this problem, I would recommend using the principles of rotational dynamics to analyze the motion of the pulley and the block. This would involve calculating the torque on the pulley and applying the equations of rotational motion to determine the angular acceleration of the pulley. From there, the linear acceleration of the block can be calculated using the relationship between linear and angular acceleration for objects connected by a rope.

Once the acceleration of the block is determined, the time to move 30cm can be calculated using the appropriate kinematic equation. This approach will take into account the changing tension in the rope and provide a more accurate answer.

It is also important to consider any other external factors, such as air resistance, that may affect the motion of the block and pulley system. These should be taken into account in the calculations to ensure accuracy. Overall, careful consideration of all factors and use of appropriate equations will lead to an accurate and scientifically sound solution to this problem.