Question involving angular acceleration of a spinning wheel

[as2528]

Homework Statement

A rotating wheel requires 3.00 s to complete 37.0 revolutions. Its angular speed at the end of the 3.00-s interval is 98.0 rad/s. What is the constant angular acceleration of the wheel?

Relevant Equations

a=(dw/dt)
w=(dtheta/dt)

wfinal=98.0 rad/s, dt=3.00s

w=(37 revs/3)=>w=(37 revs*(2*pi/1))/3=>w=77.493

a=(98-77.493)/3=>a=6.8357

My answer is exactly half of the correct answer. Where did I go wrong?

 

[PeroK]

Homework Statement:: A rotating wheel requires 3.00 s to complete 37.0 revolutions. Its angular speed at the end of the 3.00-s interval is 98.0 rad/s. What is the constant angular acceleration of the wheel?
Relevant Equations:: a=(dw/dt)
w=(dtheta/dt)

wfinal=98.0 rad/s, dt=3.00s

w=(37 revs/3)=>w=(37 revs*(2*pi/1))/3=>w=77.493

a=(98-77.493)/3=>a=6.8357

My answer is exactly half of the correct answer. Where did I go wrong?

The question seems over specified to me. If the angular acceleration is constant, then it must be $\dfrac {98}{3}$ radians per second per second.

Does that imply 37 revolutions total?

 

[PeroK]

Unless, of course, ...?

 

[as2528]

Unless, of course, ...?

Yes what I understood was 37 total revs over the 3 seconds.

 

[PeroK]

Yes what I understood was 37 total revs over the 3 seconds.

Okay. I see what you did. What is $w$ in your equations?

 

[as2528]

Okay. I see what you did. What is $w$ in your equations?

$w$ I wrote as w because I couldn't figure out how to format it.

 

[kuruman]

It is true that $\alpha=\dfrac{\omega_f-\omega_i}{\Delta t}.$

However, you used the average velocity 37/3 rev/s instead of the initial velocity, $\alpha=\dfrac{\omega_f-\omega_{\text{avg.}}}{\Delta t}.$ The initial angular velocity is assumed to be zero although not explicitly stated.

 

[as2528]

It is true that $\alpha=\dfrac{\omega_f-\omega_i}{\Delta t}.$

However, you used the average velocity 37/3 rev/s instead of the initial velocity, $\alpha=\dfrac{\omega_f-\omega_{\text{avg.}}}{\Delta t}.$ The initial angular velocity is assumed to be zero although not explicitly stated.

Got it, thanks!

 

[PeroK]

It is true that $\alpha=\dfrac{\omega_f-\omega_i}{\Delta t}.$

However, you used the average velocity 37/3 rev/s instead of the initial velocity, $\alpha=\dfrac{\omega_f-\omega_{\text{avg.}}}{\Delta t}.$ The initial angular velocity is assumed to be zero although not explicitly stated.

The initial angular velocity cannot be zero. That's what confused me initially.

As you say, the OP used average instead of initial velocity.

 

[PeroK]

Here's a variation on a theme:
$$\alpha = \frac{2\omega_f}{\Delta t} - \frac{2\theta}{(\Delta t)^2} =2\frac{\omega_f - \omega_{avg}}{\Delta t}$$

 

[haruspex]

Homework Statement:: A rotating wheel requires 3.00 s to complete 37.0 revolutions. Its angular speed at the end of the 3.00-s interval is 98.0 rad/s. What is the constant angular acceleration of the wheel?
Relevant Equations:: a=(dw/dt)
w=(dtheta/dt)

Constant angular acceleration is analogous to constant linear acceleration. You can use analogues of the standard SUVAT equations.
SUVAT is named for five standard variables (in a common formulation):
s distance
u initial speed
v final speed
a acceleration
t time
Correspondingly, there are five equations, each omitting one variable. That is, you only need to know three to determine the other two.
In the present case you are given s, v and t, and you wish to find a, so pick the equation involving those four.

Note that you do not need to assume that u=0, but it seems it is, near enough.

 

[jbriggs444]

My answer is exactly half of the correct answer. Where did I go wrong?

You have received good advice on the source of your factor of two discrepancy. It occurred in the part of the solution that you did not write down -- the reasoning process that took you from the story to the equations.

It is easy, oh so very easy, to jump straight from the story to an equation, plug in some numbers and read off an answer. [I frustrated all of my teachers by doing exactly that, even though they always begged all of us to show our danged work].

It is better to proceed as if you were explaining the problem to someone else. Which you should be -- you should be explaining it to us. Write down an explanation for why you chose the equation(s) you did and an explanation for which variable names stand for what values.

In the case at hand we have:

A rotating wheel requires 3.00 s to complete 37.0 revolutions. Its angular speed at the end of the 3.00-s interval is 98.0 rad/s. What is the constant angular acceleration of the wheel?

As I first look at the problem, I see that we have a wheel rotating 37 times in 3 seconds. But it is not a constant rotation rate. We have angular acceleration at a constant rate. We are not given an initial rotation rate.

It is immediately clear that the starting rotation rate will not be a rational number when expressed in rotations per second, so the starting rotation rate cannot be zero. You cannot subtract a rational (the average rotation rate) from an irrational (the final rotation rate when converted to rotations per second) and get a rational (the initial rotation rate). Alternately, one could have divided the final rotation rate by two to check whether it was equal to the average rate. One is rational, the other is irrational, so they are clearly unequal without even punching any numbers into a calculator. Not that there is anything wrong with punching numbers into a calculator.

The 37 rotations in 3 seconds must be the average rotation rate over that three second interval.

We also have a final rotation rate of 98.0 rad/s.

So I pause to take stock. We will have a unit conversion problem to deal with. We could work with rotations per second or radians per second. Best practice is to do the algebra first and do the unit conversions once we have a formula that needs inputs. So save the unit conversions for later.

It is time to write down an equation. What equation can we justify and write down?

We have an average rotation rate and a final rotation rate. To compute an acceleration, it would be good to have two known rotation rates and a time interval between them. We can get that...

In the case of constant acceleration, the average rotation rate is equal to the rotation rate at the midpoint of the interval. That is a handy fact to keep in your hip pocket.

We can consider the rotation rate at the midpoint of the interval as our starting rate, the rotation rate at the end of the interval as our final rotation rate and the second half of the interval as our interval of interest for calculating an acceleration.

So we can write down the generic equation for constant acceleration:$$\alpha = \frac{\omega_\text{f} - \omega_\text{avg}}{\Delta t/2} = 2\frac{\omega_\text{f} - \omega_\text{avg}}{\Delta t}$$Where $\alpha$ is our desired angular acceleration, $\omega_\text{f}$ is the given final rotation rate of 98.0 rad/s and $\omega_\text{avg}$ is the given average rotation rate of 37.0 rotations in 3.00 s.

Then we just have to convert the inputs into a set of consistent units, evaluate the formula and round to the appropriate number of significant digits.

 

[as2528]

You have received good advice on the source of your factor of two discrepancy. It occurred in the part of the solution that you did not write down -- the reasoning process that took you from the story to the equations.

It is easy, oh so very easy, to jump straight from the story to an equation, plug in some numbers and read off an answer. [I frustrated all of my teachers by doing exactly that, even though they always begged all of us to show our danged work].

It is better to proceed as if you were explaining the problem to someone else. Which you should be -- you should be explaining it to us. Write down an explanation for why you chose the equation(s) you did and an explanation for which variable names stand for what values.

In the case at hand we have:

As I first look at the problem, I see that we have a wheel rotating 37 times in 3 seconds. But it is not a constant rotation rate. We have angular acceleration at a constant rate. We are not given an initial rotation rate.

It is immediately clear that the starting rotation rate will not be a rational number when expressed in rotations per second, so the starting rotation rate cannot be zero. You cannot subtract a rational (the average rotation rate) from an irrational (the final rotation rate when converted to rotations per second) and get a rational (the initial rotation rate). Alternately, one could have divided the final rotation rate by two to check whether it was equal to the average rate. One is rational, the other is irrational, so they are clearly unequal without even punching any numbers into a calculator. Not that there is anything wrong with punching numbers into a calculator.

The 37 rotations in 3 seconds must be the average rotation rate over that three second interval.

We also have a final rotation rate of 98.0 rad/s.

So I pause to take stock. We will have a unit conversion problem to deal with. We could work with rotations per second or radians per second. Best practice is to do the algebra first and do the unit conversions once we have a formula that needs inputs. So save the unit conversions for later.

It is time to write down an equation. What equation can we justify and write down?

We have an average rotation rate and a final rotation rate. To compute an acceleration, it would be good to have two known rotation rates and a time interval between them. We can get that...

In the case of constant acceleration, the average rotation rate is equal to the rotation rate at the midpoint of the interval. That is a handy fact to keep in your hip pocket.

We can consider the rotation rate at the midpoint of the interval as our starting rate, the rotation rate at the end of the interval as our final rotation rate and the second half of the interval as our interval of interest for calculating an acceleration.

So we can write down the generic equation for constant acceleration:$$\alpha = \frac{\omega_\text{f} - \omega_\text{avg}}{\Delta t/2} = 2\frac{\omega_\text{f} - \omega_\text{avg}}{\Delta t}$$Where $\alpha$ is our desired angular acceleration, $\omega_\text{f}$ is the given final rotation rate of 98.0 rad/s and $\omega_\text{avg}$ is the given average rotation rate of 37.0 rotations in 3.00 s.

Then we just have to convert the inputs into a set of consistent units, evaluate the formula and round to the appropriate number of significant digits.

Thank you! I appreciate the detailed breakdown of how to approach the problem. It's very helpful!