Source: High school physics (Chinese)
Problem
A person of mass $m$ rides a Ferris wheel in a vertical circle of radius $R$ at a constant speed $v$.
- What is the period of the motion?
- What is the magnitude of the normal force on the person from the seat at the highest point?
- What is the magnitude of the normal force on the person from the seat at the lowest point?
[Q1] $T = 2\pi R/v$ [Q2] $F_{N, top} = m(v^2/R - g)$ [Q3] $F_{N, bot} = m(v^2/R + g)$
The period $T$ is the time for one full circle (circumference $2\pi R$) at constant speed $v$.
$$T = \frac{2\pi R}{v}$$At the highest point, both gravity ($mg$) and the normal force ($F_{N, top}$) point downwards, providing the centripetal force $F_c = mv^2/R$. Applying Newton's second law (downwards as positive):
$$F_{N, top} + mg = m\frac{v^2}{R} \implies F_{N, top} = m\left(\frac{v^2}{R} - g\right)$$At the lowest point, the normal force ($F_{N, bot}$) points upwards and gravity ($mg$) points downwards. The net force is upwards and provides the centripetal force. Applying Newton's second law (upwards as positive):
$$F_{N, bot} - mg = m\frac{v^2}{R} \implies F_{N, bot} = m\left(\frac{v^2}{R} + g\right)$$