Revolutions and radius of wheel

[rcmango]

Homework Statement

A string is wound tightly around a wheel. When the end of the string is pulled through a distance of ? cm, the wheel rotates through ? revolutions. What is the radius of the wheel?

Homework Equations

The Attempt at a Solution

i'm confused about this simple problem because i believed that displacement and revolutions were the same for theta in a kinematics problem like this: theta = 1/2(w0 + w)t

what equation would I use to solve such a problem.

 

[Integral]

All you need is the relationship between circumference and radius. You get the circumference from the length of string pulled and the number of revolutions.

 

[ogb p]

I can provide some clarification on the concepts of revolutions and radius in this problem.

Firstly, revolutions refer to the number of times an object (in this case, the wheel) completes a full circle or rotation. This is different from displacement, which is the distance an object moves from its initial position to its final position. In this problem, the end of the string being pulled through a distance of ? cm is the displacement, while the wheel rotating through ? revolutions refers to the number of times the wheel has completed a full rotation.

To solve this problem, we can use the equation for the circumference of a circle, which is C = 2πr, where r is the radius of the wheel. We can also use the relationship between linear and angular velocity, which is v = ωr, where v is the linear velocity, ω is the angular velocity, and r is the radius. We can rearrange this equation to solve for the radius: r = v/ω.

Using these equations, we can set up a system of equations to solve for the radius. For example, if the wheel rotates through 3 revolutions (n = 3) and the end of the string is pulled through a distance of 50 cm (d = 50 cm), we can set up the following equations:

C = d = 2πr
v = ωr
ω = 2πn/t

Substituting the values into the equations, we get:

50 cm = 2πr
v = (2πn/t)r
ω = (2πn/t)

We can then solve for r by substituting the values for v and ω into the first equation:

50 cm = 2π(2πn/t)r
r = 50 cm / (2π(2πn/t))
r = 25 cm / (πn/t)

Therefore, the radius of the wheel is 25 cm / (πn/t), where n is the number of revolutions and t is the time it takes for the wheel to complete those revolutions.

In conclusion, the concept of revolutions and radius are different and can be solved using equations related to linear and angular velocity.