Rotating mass on a frictionless table pulling up an object.

[vvanessa]

Homework Statement

(General Physics - Algebra based)
A 0.75 kg puck rotates in a horizontal circular motion on a frictionless table. It is attached to a string that is looped through a hole at the center of table and circular path. Attached to the other end of the string is a 1.5 kg block that is pulled up at a rate of 3 m/s². What is velocity of the puck?

Knowns
m₁ = 0.75 kg
m₂ = 1.5 kg
a = 3 m/s²
g = 9.8 m/s²

Unknowns
v (tangential velocity of puck)

Homework Equations

F_w2 = m₂g
a_c = v²/r

The Attempt at a Solution

∑F_x1 = T = m₁a_c
⇒ T = m₁v²/r

∑F_y2 = T - F_w = m₂a
⇒ (m₁v²/r) - m₂g = m₂a
⇒v = √[m₂r(a+g)/m₁] = √([25.6r m/s²)

How can I find the value of v w/o knowing the radius?

Also, if the block is being pulled up, then doesn't that mean the radius of the circular path where the puck is attached is increasing and, thus, v is increasing over time?

 

[Baluncore]

Welcome to PF.

A 0.75 kg puck rotates in a horizontal circular motion on a frictionless table. It is attached to a string that is looped through a hole at the center of table and circular path. Attached to the other end of the string is a 1.5 kg block that is pulled up at a rate of 3 m/s2.

There are too many interpretations of the situation. Is there a diagram that shows what is meant by "horizontal circular motion" of "puck on a frictionless table". Does that mean a vertical axis of rotation, which way through the puck?

 

[jbriggs444]

There are too many interpretations of the situation.

Agreed. As I read the problem, it is self-contradictory.

If the string is moving through the hole, the puck's path will not be in a circle centered on the hole. It would instead be at an angle, not even tangent to such a circle.

If the string were momentarily motionless but accelerating through the hole, the puck's path would still not be in a circle centered on the hole. The path would be tangent, but the curvature would be wrong. The momentary center of curvature would not be at the position of the hole.

The problem statement both requires and forbids the string to be accelerating through the hole.

 

[vvanessa]

It's one of these types of problems:

 

[Baluncore]

Also, if the block is being pulled up, then doesn't that mean the radius of the circular path where the puck is attached is increasing and, thus, v is increasing over time?

The puck 'orbits' the hole. As the puck moves further from the hole in a spiral path it travels more slowly, until the process reverses and the weight starts to fall, the puck never stops, the radius stops increasing, then begins to decrease, so it continues to orbit in the same sense. The masses are related only by the length and tension in the string.
Linear and angular conservation of energy and momentum hold in the system.

 

[haruspex]

It's one of these types of problems:
View attachment 214874

As @jbriggs444 posted, the question is broken. The puck would not move in a circle. The differential equation looks nasty.
It s hard even to guess what invalid approach the questioner is expecting you to use.

 

[haruspex]

Linear and angular conservation of energy and momentum hold in the system.

Energy and angular momentum, yes, but linear?