Question

An object with a mass of `3 kg` is traveling in a circular path of a radius of `2 m`. If the object's angular velocity changes from `1 Hz` to `8 Hz` in `5 s`, what torque was applied to the object?

Answer 1

It depends on whether it's a vertical loop or a horizontal loop, but let's assume it's horizontal and imagine a so-called "massless" string attached to the object.

Let's try drawing a Free-Body diagram for this:

Now, we can use a simple kinematics equation analog relating `Deltavecomega` with `vecalpha`.

Just like how:

`vecv_f = vecv_i + vecat`

... we have:

`\mathbf(vecomega_f = vecomega_i + vecalphat)`

So with this, we can find `alpha`, and with `alpha`, we can input that into this equation for the sum of the torques:

`\mathbf(sumtau = Ialpha)`

Next, we need the inertia for a mass rotating about an axis. It should be `I = kmr^2`, where `k = 1`. So:

`sumtau_"ext"`

`= color(green)(I)color(highlight)(alpha)`

`= color(green)(mr^2)color(highlight)(((vecomega_f - vecomega_i)/t))`

`= color(green)(("3 kg")("2 m")^2)color(highlight)((("8 rad/s" - "1 rad/s")/"5 s"))`

`= color(blue)("16.8 N"cdot"m")`