Speed of a solid cylinder after unwinding

In summary, the problem asks for the angular velocity of a cylinder as it descends and the student was not able to find a formula due to missing variables.
  • #1
BMZ
3
0

Homework Statement


The problem states the following:

One end of a string is attached to the ceiling while the other end is wrapped around a solid cylinder of mass 0.20 kg and radius 0.030m. The cylinder is released from rest and the string unwinds as the cylinder rotates and accelerates downward. Determine the a) Solid cylinder's speed after it has traveled 1.0m downward and b) its rate of acceleration.

So given variables are
mass = 0.20kg
radius of cylinder = 0.030m
and a displacement of 1.0m downwards.

Homework Equations



I can't seem to find the correct one for this. I have a sheet and my Physics book with all the formulas yet all the ones I've found up to now require me to have the time in order to find the speed. I have been trying for a while now, and can't seem to find anything relevant to this. Only closest thing I could find would be rotational Kinetic Energy which would be

KEr = ½Iω2, but then there would be two unknowns at the end of this which would be ω and the KEr

If I had Δt then
Θ = ω0t + ½αt2 could be used I think. Maybe there's an error in the question, which is quite possible since my teacher makes mistakes, but unlikely.

The Attempt at a Solution



I haven't been able to attempt it at all since all the formulas that I can remember off the top of my head and the sheets seem to be missing variables. A formula would push me in the right direction, but I've looked everywhere for formulas regarding Angular displacement, and Angular velocity or even linear velocity/displacement but I seem to be missing variables like in:

ω = Θ/t
ω = v⋅r
--
Thank you for any help.
 
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  • #2
..there would be two unknowns at the end of this which would be ω and the KEr

I haven't tried to solve the problem but perhaps it would help to ask yourself..

Where does the KE come from?
The string unwinds so is there is a relationship between rotation and some other parameter?
 
  • #3
CWatters said:
I haven't tried to solve the problem but perhaps it would help to ask yourself..

Where does the KE come from?
The string unwinds so is there is a relationship between rotation and some other parameter?

Well, I think the KE would come from the rotation of the cylinder as it unwinds from the string. But at the beginning it doesn't have any if it's not moving right? Then maybe the left side would be zero since the rotational kinetic energy is zero at the beginning, then that would be a solvable equation but I don't think it's like that.

All the relationships I can think of relating to the rotation is the radius, the velocity and maybe the length of the string. But it says "after it has traveled 1.0m downwards", doesn't specify whether the string ends there but I don't think knowing that the string is 1m long would help much.

I thought about using Potential Energy = mgh, but I don't really know H, unless the 1m is H and it can be used there.
 
  • #4
BMZ said:
I thought about using Potential Energy = mgh, but I don't really know H, unless the 1m is H and it can be used there.
In mgh, h is the change in height. The change in height is indeed 1m.
The formula you quote for rotational KE is correct, but the cylinder is not simply rotating about its central axis. There are two ways to handle this, and I don't know which you may be familiar with. You can treat the cylinder's motion as a rotation about its centre, plus a linear motion, or as a rotation about a different axis, with no linear motion. Your choice.
 
  • #5
BMZ said:
Well, I think the KE would come from the rotation of the cylinder as it unwinds from the string. But at the beginning it doesn't have any if it's not moving right?
The KE doesn't "come from the rotation of the cylinder" that's where some of it goes. The cylinder is also descending with some velocity as well so also it has linear KE.

BMZ said:
I thought about using Potential Energy = mgh, but I don't really know H, unless the 1m is H and it can be used there.
Why not?

BMZ said:
All the relationships I can think of relating to the rotation is the radius, the velocity and maybe the length of the string.

That's the right track. There is a relationship between the length of string unwound and the number of revolutions of the cylinder. There is also a relationship between the velocity of the string relative to the cylinder and the angular velocity of the cylinder (just as there is between the angular velocity of a car wheel and the velocity of the car/road)

Cross posted with haruspex.
 
  • #6
Well, if Potential Energy fits the problem by using the 1m change in height in mgh, then what would come to mind is

Potential Energy = Linear KE + Angular KE
which would be

mgh = ½mv2 + ½Iω2

Then I = 0.5mr2 since it's a solid cylinder.

equation would then be mgh = ½mv2 + ½(½mr22

You would cancel out all the masses, and since v = r⋅ω, this would end being.

gh = ½v2 + ¼v2

which by combining like terms you end up with

gh = ¾v2

But I'm not entirely sure whether that would be correct for this.
 
  • #7
BMZ said:
gh = ¾v2
Work the numbers, then compare to the velocity of a simple free-falling object.
 

1. What is the concept of speed of a solid cylinder after unwinding?

The speed of a solid cylinder after unwinding refers to the rate at which the cylinder rotates after it has been unwound from a stationary position. It is a measurement of the angular velocity of the cylinder, or how fast it is spinning.

2. How is the speed of a solid cylinder after unwinding calculated?

The speed of a solid cylinder after unwinding can be calculated using the equation v = ωr, where v is the linear speed, ω is the angular velocity, and r is the radius of the cylinder. This equation is derived from the relationship between linear and angular velocity, v = rω.

3. What factors affect the speed of a solid cylinder after unwinding?

The speed of a solid cylinder after unwinding can be affected by several factors, including the initial angular velocity of the cylinder, the mass and distribution of the mass on the cylinder, the surface friction, and the radius of the cylinder. In addition, external forces such as air resistance or gravity can also impact the speed of the cylinder.

4. How does the speed of a solid cylinder after unwinding relate to its moment of inertia?

The moment of inertia, which is a measure of an object's resistance to changes in its rotational motion, is directly proportional to the speed of a solid cylinder after unwinding. This means that the greater the moment of inertia, the lower the speed of the cylinder will be.

5. Can the speed of a solid cylinder after unwinding be changed?

Yes, the speed of a solid cylinder after unwinding can be changed by altering the factors that affect it, such as the initial angular velocity or the mass distribution on the cylinder. Additionally, external forces can also be applied to change the speed of the cylinder. For example, applying a torque can increase or decrease the speed of the cylinder.

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