Moment of inertia and work problem

In summary, the question asks for the amount of work done on a spool with a radius of 0.500 m and a mass of 1.00 kg when a 4.00-m length of nylon cord is pulled from it with a constant acceleration of 2.82 m/s2. The answer can be found using either the formula for tension or conservation of energy, with the correct answer being 7.896 joules. The formula for moment of inertia for a solid cylinder should be used, rather than the one for a hollow cylinder, and the correct value for the polar moment of inertia is mr^2.
  • #1
coolbyte
4
0

Homework Statement



A 4.00-m length of light nylon cord is wound around a uniform cylindrical spool of radius 0.500 m and mass 1.00 kg. The spool is mounted on a frictionless axle and is initially at rest. The cord is pulled from the spool with a constant acceleration of magnitude 2.82 m/s2.

How much work has been done on the spool when it reaches an angular speed of 7.95 rad/s?


Homework Equations



Angular acceleration = a/r
Moment of inertia of the cylinder is mr^2

The Attempt at a Solution



I solved for tension first: T*0.5 = 1*(0.5)^2*(2.82 / 0.5)=> So T=2.82 N

Then using kinematic equations I solved for theta:

Angular acceleration = a/r = 2.82/0.5 = 5.64

so

7.95^2 = 2 * 5.64 * theta ==> theta = 5.6

so work = 2.82 * 5.6 * 0.5 = 7.896 joules
However webassign says I'm wrong, any help would be greatly appreciated as I spent too much time into this, thanks.
 
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  • #2
coolbyte said:
Moment of inertia of the cylinder is mr^2
That would be for a hollow cylinder. I would assume the spool is solid.
I solved for tension first: T*0.5 = 1*(0.5)^2*(2.82 / 0.5)=> So T=2.82 N
That's a long way round. Just use the angular KE formula, Iω2/2.
 
  • #3
It looks like you calculated the polar moment of inertia incorrectly. Recheck your formula. Also, even though your approach should deliver the correct answer, it would be much easier to solve this problem by applying conservation of energy. That way, you wouldn't even have to calculate the tension.
 
  • #4
haruspex said:
That would be for a hollow cylinder. I would assume the spool is solid.

That's a long way round. Just use the angular KE formula, Iω2/2.

Thanks mate, you're right, those formulas always confuse me.
 
  • #5
coolbyte said:
Thanks mate, you're right, those formulas always confuse me.
mr2 applies for a point mass at distance r, so that must also be the formula for a hoop about a line orthogonal to the plane of the hoop, and for a hollow thin shell cylinder about its axis. That's because in each of those cases every part part of the body rotates at radius r from the axis. For a solid disc or cylinder, or for a sphere (solid or hollow), or for a hoop rotating about a diameter, it must be less, since no part rotates at distance greater than r but some parts rotate at a shorter distance.
 

1. What is moment of inertia?

Moment of inertia is a physical property of a rotating object that describes its resistance to changes in its rotational motion. It is similar to mass in linear motion and is dependent on an object's mass and distribution of mass relative to its axis of rotation.

2. How is moment of inertia calculated?

The moment of inertia of a point mass is the product of the mass and the square of its distance from the axis of rotation. For a continuous object, it can be calculated by integrating the mass of each infinitesimal element multiplied by its distance from the axis squared.

3. What is the relationship between moment of inertia and work?

Moment of inertia is related to work through the concept of rotational kinetic energy. Work is the transfer of energy, and in rotational motion, the energy is stored as rotational kinetic energy, which is proportional to the moment of inertia. Therefore, the larger the moment of inertia, the more work is required to change the rotational motion of an object.

4. Can moment of inertia be negative?

No, moment of inertia cannot be negative. It is a physical property that represents an object's resistance to changes in its rotational motion. Therefore, it must always be positive or zero.

5. How does the distribution of mass affect the moment of inertia?

The distribution of mass has a significant impact on the moment of inertia. Objects with more mass concentrated farther from the axis of rotation have a larger moment of inertia than objects with the same mass but a more centralized distribution. This is because the distance from the axis of rotation plays a crucial role in the calculation of moment of inertia.

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