# Determine the Velocity and Magnitude of Acceleration on the Wheel

• Northbysouth
In summary: Consequently, the magnitude of the acceleration is simply the magnitude of the radial acceleration:a = rθ'
Northbysouth
1.

A bicycle is placed on a service rack with its wheels hanging free. As part of a bearing test, the front wheel is spun at the rate of N = 59 rev/min. Assume that this rate is constant, and determine the speed v and magnitude a of the acceleration of point A.

I have attached an image of the question.

## Homework Equations

v = ds/dt = ρ dβ/dt

v = rθ'

## The Attempt at a Solution

I initially tried converting N = 59 rev/min to 6.178466 rad/sec to use in:

v = ds/dt = ρ dβ/dt

I suspect that I need to use v = rθ' but I'm not sure how you take the derivative of an angle. Help is appreciated.

#### Attachments

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Northbysouth said:
I suspect that I need to use v = rθ' but I'm not sure how you take the derivative of an angle.

You don't have to. The angular speed ω = dθ/dt is given in the problem. It's 59 rpm.

By the way, why didn't whoever made this assignment make it 60 rpm, or one rotation per second, which is 2pi radians per second, so that the numbers would work out nicely? Just to be perverse? EDIT: This was a rhetorical question, in case it wasn't clear.

Last edited:
What are the meaning and the units for the symbols ρ and β? How does the equation v = ρ dβ/dt differ from the equation v = rθ'?

So, I think I have the velocity.

I took the angular acceleration N = 59 rev/min and converted it to 6.178466 rad/sec wfter which I took the 26" diameter and:

26 in/12 = 2.1667 ft (diameter

r = 1.0833

v = 6.6933 ft/sec

I'm a little confused with the acceleration, when it asks for the magnitude of the acceleration does that mean I have to calculate the tangential and normal acceleration and then take the magnitude?

Point A on the rim of the wheel moves at a constant speed, which means that it is in uniform circular motion. There is no tangential acceleration, only radial (centripetal) acceleration.

## 1. What is the formula for determining velocity and magnitude of acceleration on a wheel?

The formula for determining velocity is v = d/t, where v is velocity, d is distance, and t is time. The formula for determining magnitude of acceleration is a = (vf - vi)/t, where a is acceleration, vf is final velocity, vi is initial velocity, and t is time.

## 2. How do you measure the distance traveled by a wheel?

The distance traveled by a wheel can be measured using a ruler or tape measure. Place the measuring tool at the starting point of the wheel and measure the distance to the ending point. This will give you the distance traveled by the wheel.

## 3. Can the velocity and magnitude of acceleration on a wheel change?

Yes, the velocity and magnitude of acceleration on a wheel can change depending on various factors such as the force applied, the surface it is rolling on, and the weight of the wheel.

## 4. What is the unit of measurement for velocity and magnitude of acceleration?

The unit of measurement for velocity is meters per second (m/s) and for magnitude of acceleration is meters per second squared (m/s^2).

## 5. How can the velocity and magnitude of acceleration on a wheel be used in real-life applications?

The velocity and magnitude of acceleration on a wheel are important in various real-life applications such as designing vehicles, understanding the movement of objects in motion, and developing technologies for sports and amusement rides. They are also essential in studying the motion of celestial bodies and spacecrafts.

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