Nonuniform Circular Motion - Find Maximum Total Acceleration

[LovePhys]

Homework Statement

A turntable is rotating at a constant angular velocity of ω = 4.0 rad/s in the direction of a clockwise fashion. There is a ten-cent coin on the turntable, at a distance of 5 cm from the axis of rotation.

Consider the time interval during which the turntable is accelerated initially from rest to its final angular velocity (ωf = 4.0 rad/s) . This is achieved with a constant angular acceleration (α) for 0.5 s.

Find an expression for the magnitude of total acceleration of the coin in terms of α and ω and use this to determine the maximum total acceleration experienced by the coin.

Homework Equations

[itex]a=\sqrt{a_{r}^2+a_{t}^2}[/itex]
[itex]a_{r}=ω^2r[/itex]
[itex]a_{t}=αr[/itex]

The Attempt at a Solution

Using those equations, I find an expression for the total acceleration:
[itex]a=r\sqrt{ω^4+α^2}[/itex]
I don't understand why the problem is asking for a maximum total acceleration. I am already given the values of r, ω, and α, can I just substitute them into find the answer? Another thing that came into my mind when the word "maximum" pops up is finding the derivative of a function of a(t) with respect to time, but apparently I do not have that function here...

Any help would be greatly appreciated.
LovePhys

 

[Doc Al]

I don't understand why the problem is asking for a maximum total acceleration. I am already given the values of r, ω, and α, can I just substitute them into find the answer?

Well, since ω varies over the given interval, you have to determine its maximum value. But I suspect you can handle it.

 

[LovePhys]

@Doc Al: Thanks for the hint. I think the maximum angular velocity is also the final angular velocity ω=4rad/s, isn't it?
[itex]α=\frac{Δω}{Δt}=\frac{4}{0.5}=8 (rad/s^2) [/itex]
So finally maximum [itex] a=0.05\sqrt{4^4+8^2}≈0.894(m/s^2) [/itex]

 

[Doc Al]

@Doc Al: Thanks for the hint. I think the maximum angular velocity is also the final angular velocity ω=4rad/s, isn't it?

Exactly.

[itex]α=\frac{Δω}{Δt}=\frac{4}{0.5}=8 (rad/s^2) [/itex]
So finally maximum [itex] a=0.05\sqrt{4^4+8^2}≈0.894(m/s^2) [/itex]

Looks good to me.

 

[Nickie]

icist

As a scientist, your response should be focused on the physics principles involved in solving this problem rather than simply giving the solution. Here is a possible response:

The problem is asking for the maximum total acceleration experienced by the coin during the initial acceleration period. This is important because it tells us the maximum force that the coin will experience, which can be useful in understanding the stability of the coin on the turntable.

To solve this problem, we can use the given equations for total acceleration and the known values of ω and α. However, it is important to note that the total acceleration is a vector quantity, and we need to consider both the radial and tangential components of acceleration.

Using the equation for radial acceleration, we can see that it is directly proportional to ω^2, which means that as the angular velocity increases, the radial acceleration also increases. Similarly, the tangential acceleration is directly proportional to the angular acceleration α, so as the turntable accelerates faster, the tangential acceleration also increases.

To find the maximum total acceleration, we can substitute the known values of ω and α into the equation for total acceleration and solve for the maximum value. This will give us the maximum force that the coin will experience during the initial acceleration period.

Furthermore, we can also plot the total acceleration as a function of time and take the derivative to find the maximum value, as you suggested. This would give us a more detailed understanding of how the acceleration changes over time and at what point it reaches its maximum.

In conclusion, the maximum total acceleration experienced by the coin on the turntable can be found by using the given equations and known values of ω and α. It is important to consider both the radial and tangential components of acceleration and to understand how they change as the turntable accelerates.