Determine the Velocity and Magnitude of Acceleration on the Wheel

[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

but this left me with radians/sec instead of ft/sec.

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.

 

[cepheid]

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.

 

[TSny]

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'?

 

[Northbysouth]

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 = (1.0833 ft)(6.178466 rad/sec)

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?

 

[cepheid]

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.