The radius and centripetal acceleration of a bobsled turn on an ideally banked curve

Problem:

(a) What is the radius of a bobsled turn banked at 75.0° and taken at 30.0 m/s, assuming it is ideally banked?

(b) Calculate the centripetal acceleration.

(c) Does this acceleration seem large to you?

Solution:

Part A

For ideally banked curved, the ideal banking angle is given by the formula \displaystyle \tan \theta = \frac{v^2}{rg}. We can solve for r in terms of all the other variables, and we should come up with

r = \frac{v^2}{g \tan \theta}

We are given the following values:

• ideal banking angle, \displaystyle \theta = 75.0\ ^\circ

• linear speed, \displaystyle v=30.0\ \text{m/s}

• acceleration due to gravity, \displaystyle g=9.81\ \text{m/s}^2

If we substitute all the given values into our formula for r, we have

\begin{align*}
r & = \frac{v^2}{g \tan \theta} \\ \\
r & = \frac{\left( 30.0\ \text{m/s} \right)^2}{\left( 9.81\ \text{m/s}^2 \right)\left( \tan 75^\circ  \right)} \\ \\
r & = 24.5825\ \text{m} \\ \\
r & = 24.6\ \text{m} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

The radius of the ideally banked curve is approximately 24.6\ \text{m}.

Part B

The centripetal acceleration a_c can be solved using the formula

a_c = \frac{v^2}{r}

Substituting the given values, we have

\begin{align*}
a_c & = \frac{v^2}{r} \\ \\
a_c & = \frac{\left( 30.0\ \text{m/s} \right)^2}{24.5825\ \text{m}} \\ \\
a_c & = 36.6114\ \text{m/s}^2 \\ \\
a_c & = 36.6 \ \text{m/s}^2 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*} 

The centripetal acceleration is about 36.6\ \text{m/s}^2.

Part C

To know how large is the computed centripetal acceleration, we can compare it with that of acceleration due to gravity.

\frac{a_c}{g} = \frac{36.6114\ \text{m/s}^2}{9.81\ \text{m/s}^2} = 3.73

The computed centripetal acceleration is 3.73 times the acceleration due to gravity. That is a_c = 3.73g.

This does not seem too large, but it is clear that bobsledders feel a lot of force on
them going through sharply banked turns!