Question

A model train, with a mass of `4 kg`, is moving on a circular track with a radius of `8 m`. If the train's rate of revolution changes from `1/6 Hz` to `1/2 Hz`, by how much will the centripetal force applied by the tracks change by?

Answer 1

The change in centripetal force will be `DeltaF_c=256pi/9`N or `DeltaF_c = 280.73N`

Explanation:

The problem describes a train traveling on a circular track that starts at an initial speed, but then speeds up to a final speed (that is, it increases its rate of revolution).

A greater centripetal force will be required at the final rate of revolution, `w_f`, compared to the initial rate of revolution (or angular velocity), `w_i`, in order to keep the train traveling on a circular path (and not fly off the tracks!!). They want us to find the difference between these two centripetal forces.

We can express this difference mathematically as

1) `Delta F_c = F_(c_f) - F_(c_i)`

here `F_(c_f)` and `F_(c_i)` are the final and initial centripetal forces on the train respectively.

In general, the formula for centripetal force can be expressed as

2) `F_c = m*v^2/r`

here `v` represents the tangential velocity of an object (having a mass `m`) traveling around a circle of radius `r`. `F_c` is of course the centripetal force.

Centripetal force, `F_c`, can also be expressed as

3) `F_c = m*w^2*r` (why? because `v=w*r`!)

where `w` is the angular velocity of the object (and all the other variables, i.e. `F_c, m, and r`, represent the same physical quantities as before). These formulas for centripetal force, `F_c`, are equivalent, but using the second form makes the solution a bit more straight forward.

Ok, keeping this more useful form for `F_c` in mind, let’s express equation 1) in more detail...

4) `Delta F_c = F_(c_f) - F_(c_i) = m*w_f^2*r- m*w_i^2*r`

here the final rate of revolution is `w_f=1/2Hz` and initial rate of revolution is `w_i=1/6Hz`.

Simplifying further, with a little more algebra (i.e. taking advantage of the distributive property).

5) `Delta F_c = F_(c_f) - F_(c_i) = m*r(w_f^2- w_i^2)`

It’s given that the mass of the train is `m = 4kg` and the radius of the circular track is `8m`. Noting that 1 revolution equals `2pi` radians, and substituting, we get

6)`Delta F_c = m*r(w_f^2- w_i^2)=4kg*8m*(((2*pi)/2)^2- ((2*pi)/6)^2)`

`Delta F_c =280.73N`