Example Engineering/Physics question

This question seeks to discover how well an applicant can link various concepts from Physics/ Mechanics to solve a problem. It also tests some basic mathematical skills.

The ultimate question to which an answer is sought is:

From what height H must a rollercoaster car be released if it is to successfully travel around a loop of radius R without falling off at the top of the loop?

The discussion at interview breaks this down into a sequence of questions: having satisfactorily resolved one, perhaps with some help and prompting from the interviewer, we move on to the next.

1.   Assuming frictional forces can be neglected, if the car is released from rest at height H, how fast will it be moving when it reaches point A?

This can, perhaps with a hint from the interviewer, be answered using the principle of the conservation of energy – potential energy is converted into kinetic energy as the car descends the track:

where g is the acceleration due to gravity.

2.   If the car follows the track to the top of the loop at B what then is its speed?

This too can be answered using the principle of the conservation of energy. A very alert applicant will realise they can use the previous result but replacing H with the net change of height from the start to B :

3.   When the car is at B what forces are acting on it and in what directions?

Here we are looking to see how well the applicant understands the physical origin of forces. If frictional forces are still neglected, the only forces acting are the car’s weight and the reaction from the track on the car – both of which act straight downwards.

4.   What is the acceleration of the car at B and in what direction?

The applicant will hopefully recognise that this is an example of circular motion. If they cannot remember the expression for centripetal acceleration

we would tell them this result (without penalty). If the applicant has not yet covered circular motion (rare) we would briefly explain the key points needed to make progress on the question.

5.   How are the forces acting on the car at B and its acceleration linked?

This requires the applicant to recognise that Newton’s Second Law  can be applied to this situation. If all goes well, they will reach the result:

in which N is the reaction from the track on the car and M is the car’s mass.

6.   Under what circumstances will the car lose contact with the track at B?

Here we hope the applicant will recognise that the car will lose contact if the reaction N falls to zero, which will happen if the car is not going fast enough at that point.

7.   So, from what height must the car be released if this is not to happen?

Pulling together the preceding analysis the applicant can show that  if the car is not to lose contact with the track at B.

8. If an applicant does all of the above well and quickly then the question can be extended further by asking them to think qualitatively about how the situation changes if the object on the rollercoaster track is a ball bearing rather than a car, so it gains both translational and rotational kinetic energy as it descends the track.