Question

An object with a mass of `2 kg` is traveling in a circular path of a radius of `4 m`. If the object's angular velocity changes from `3 Hz` to `8 Hz` in `2 s`, what torque was applied to the object?

Answer 1

`tau=503"Nm"`

Explanation:

I will assume that you meant that the frequency changes from `3"Hz"` to `8"Hz"` (which in turn will affect the angular velocity).

Torque can be expressed as the rate of change of angular momentum.

`(1)" "color(darkblue)(tau=(dL)/(dt))`

Angular momentum can be described by the equation:

`L=Iomega`

where `I` is the moment of inertia and `omega` is the angular velocity

Substituting this into equation `(1)`, we have:

`(2)" "tau=(d(Iomega))/(dt)`

The angular velocity an be expressed by the equation:

`omega=2pif`

Substituting this into equation `3`, we get:

`(3)" "tau=(d(I*2pif))/(dt)`

Because we are given that the frequency of the motion is changing we can move `I` and `2pi` outside of the differential as constants and rewrite equation `(3)` as:

`(4)" "tau=2piI*(df)/(dt)`

Finally, for a (`~~`point) mass traveling a circular path about a center axis, the moment of inertia is given by `I=mr^2`.

Additionally, we will use the average rate of change, meaning that

`(df)/(dt)=(Deltaf)/(Deltat)=(f_f-f_i)/(t_f-t_i)`.

Substituting both of the above into equation `(4)`, we get:

`(5)" "color(darkblue)(tau=2pimr^2*(f_f-f_i)/(t_f-t_i)`

We are given the following information:

• `|->m=2"kg"`
• `|->r=4"m"`
• `|->t=2"s"`
• `|->f_i=3"s"^-1`
• `|->f_f=8"s"^-1`

Substituting these values into equation `(5)`, we get:

`tau=2pi(2"kg")(4"m")^2*(8"s"^-1-3"s"^-1)/(2"s"-0"s")`

`=>=64pi"kgm"^2*5/2s^-2`

`=>=160pi"Nm"`

`=>502.655"Nm"`

`:.tau~~503"Nm"`