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#