# 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?

##### 1 Answer

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

Just like how:

#vecv_f = vecv_i + vecat#

... we have:

#\mathbf(vecomega_f = vecomega_i + vecalphat)#

So with this, we can find **the sum of the torques**:

#\mathbf(sumtau = Ialpha)#

Next, we need the inertia for **a mass rotating about an axis**. It should be

#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")#