Fairground rides

Apart from being safe fairground rides are designed to be exciting. To do this the people on it must accelerate and this is often achieved by spinning them in a circle - better still if the radius of the circle can be made to change - then their acceleration changes.

1. Roller coaster

A car on a big dipper descends from rest at a height h and then loops the loop in a circular section of track, which is all assumed to be frictionless. At the bottom of the hill it will be moving at speed v where v = (2gh)^1/2 . (You can prove this by equating kinetic energy gained to potential energy lost).

It must be moving at a velocity u at the top of the loop where the centripetal acceleration is just equal to g otherwise it will fall off.
So g = u²/r
Using the law of the conservation of energy:
Kinetic energy at top of loop = kinetic energy at the bottom - potential energy needed to reach the top of the loop:
˝ mu² = ˝ mv² - mg2r = ˝m2gh - mg2r = = ˝mrg
Therefore: mgh - 2rmg = ˝ mgr so h = 2.5mgr/mg = 2.5r

Minimum height of start of the ride for a loop of radius r = 2.5 r

2. Roundabout (carousel) – suspended chairs

A fairground ride consists of a number of chairs suspended from wires that swing out as the centre rotates faster and faster. But do the empty chairs have a bigger orbit radius than the full chairs and is the angle that the wires make with the vertical different depending on the mass of the chairs.

Let the angle of the wire to the vertical be q for a certain chair, the tension in the wire T, the mass of the chair be m, the radius of the circle be r and the speed of the chair be v.

The forces on the chair are:

Vertically: mg = Tcosq and horizontally mv²/r = T sinq
therefore tanq = sinq/cosq = [mv²/r]/mg = v²/rg

This formula shows that the angle and the radius depend on the velocity alone and not the mass of the chairs. A good job too if you think about an actual ride - if this wasn't true the wires would all get tangled up if the people in the chairs had different masses.

See also:
Motion in a vertical circle
Circular motion