Revolutions and radius of wheel

In summary, the problem involves a string being wound around a wheel, and when the end of the string is pulled a certain distance, the wheel rotates a certain number of revolutions. To find the radius of the wheel, you need to use the relationship between circumference and radius. The circumference can be found from the length of string pulled and the number of revolutions.
  • #1
rcmango
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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.
 
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  • #2
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.
 
  • #3


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.
 

1. What is the relationship between revolutions and the radius of a wheel?

The number of revolutions a wheel makes is directly proportional to the radius of the wheel. This means that as the radius of the wheel increases, the number of revolutions required to cover a given distance also increases.

2. How does the radius of a wheel affect its speed?

The larger the radius of a wheel, the faster it can travel. This is because a larger radius covers more distance in one revolution compared to a smaller radius. This means that the wheel requires fewer revolutions to cover the same distance, resulting in a higher speed.

3. Can the revolutions of a wheel affect its stability?

Yes, the revolutions of a wheel can affect its stability. If a wheel is rotating too quickly, it can become unstable and lose traction, causing it to slip or even tip over. This is why it is important to consider the appropriate number of revolutions for a given wheel size and surface conditions.

4. How does the number of revolutions affect the distance traveled by a wheel?

The number of revolutions directly affects the distance traveled by a wheel. The distance traveled is equal to the circumference of the wheel multiplied by the number of revolutions. This means that the more revolutions a wheel makes, the greater the distance it will travel.

5. How can the radius of a wheel affect the force required to turn it?

The radius of a wheel affects the amount of torque required to turn it. A larger radius means a greater distance from the center of the wheel to the point where force is applied, resulting in a greater leverage and less force needed to turn the wheel. On the other hand, a smaller radius requires more force to turn the wheel due to the shorter distance from the center to the point of force application.

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