A wheel with radius 5 cm is being pushed up a ramp at a rate of 6 cm per second. The ramp is 540 cm long, and 260 cm tall at the end. A point P is marked on the circle as shown (picture is not to scale).

Question

wheelonramp.gif

1. Traveling at 8 cm/sec, how long does it take the wheel to complete one rotation?

2. What is the initial height of the center of wheel?

3. Traveling at 5 cm/sec, in one second, how much does the height of the center of wheel increase?

4. Write an equation for the height of the center of the wheel as a function of t, time in seconds.

5. Write an equation for the height of the point P above the ground as a function of t, time in seconds.

Answer 1

It's kind of an oddly worded question - it states a speed of 6cm/sec, but goes on immediately in part 1. to change this to 8cm/s...

1. A point on the wheel travels a distance of the circumference of the wheel in one rotation, or 2π * 5 = 10π cm
10π cm divided by 8 cm/s = _____s

2. You should be able to work this out from the diagram (which can't be seen on this forum). As a hint, if the wheel were on level ground, the height of the centre of the wheel above the ground would be the wheel's radius.

3. The centre of the wheel travels 5 cm in 1 second, in a direction parallel to the ramp's surface. The angle the ramp makes with the ground is θ = tan-1(ramp slope).
So the vertical component of the centre of the wheel's displacement (the height increase) would be 5sinθ cm.

This should be enough info to get you going on the rest of the question.