Question

A roller coaster car is going over the top of a 12-m-radius circular rise. At the top of the hill, the passengers "feel light," with an apparent weight only 60% of their true weight.

How fast is the coaster moving?

Answer 1

Concepts and reason

The concepts required to solve this question are centripetal force and apparent weight.

First, determine the net force at the top of the roller coaster. Then, equate the net force with the apparent weight of the passengers.

Finally, substitute the values in the force equation to find the speed of the roller coaster.

Fundamentals

Centripetal force is the force that causes an object to move along a circular path. The expression for centripetal force is:

$F = \frac{{m{v^2}}}{r}$

Here, m is the mass of the object, v is the velocity of the object, and r is the radius of the circular path.

Weight of an object is the force on an object due to gravity. Force acting on an object due to its weight is,

$w = mg$

Here, m is the mass of the object and g is acceleration due to gravity.

The sum of all forces acting on the top of the circular motion is the equal to the net force acting on the object in the circular path which enables the object to move in the circle.

The sum of the forces acting on the passengers when they at the top of the roller coaster is as follows:

$\Sigma F = mg - N$

Here, N is the normal force acting on the passengers in the upwards direction and mg is the weight of the passengers acting downwards.

The net force which is responsible for the circular motion of the roller coaster is the centripetal force and the sum of all the forces acting on the top of the roller coaster is equal to this centripetal force.

$\Sigma F = \frac{{m{v^2}}}{r}$

Substitute $mg - N$ for $\Sigma F$ in the above expression and solve for N.

$\begin{array}{c}\\mg - N = \frac{{m{v^2}}}{r}\\\\N = mg - \frac{{m{v^2}}}{r}\\\end{array}$

The apparent weight of the passengers is 60% of their true weight. This means that the normal force acting on the passengers is equal to the 60% of the weight. The normal force acting on the passengers is as follows:

$\begin{array}{c}\\N = \left( {\frac{{60}}{{100}}} \right)mg\\\\ = 0.6mg\\\end{array}$

The expression for the normal force is as follows:

$N = mg - \frac{{m{v^2}}}{r}$

Substitute 0.6mg for N and rearrange for v.

$\begin{array}{c}\\0.6mg = mg - \frac{{m{v^2}}}{r}\\\\\frac{{{v^2}}}{r} = g\left( {1 - 0.6} \right)\\\\v = \sqrt {0.4rg} \\\end{array}$

Substitute $9.8{\rm{ m/}}{{\rm{s}}^2}$ for g, and 12.0 m for r in the above equation.

$\begin{array}{c}\\v = \sqrt {0.4\left( {12.0{\rm{ m}}} \right)\left( {9.8{\rm{ m/}}{{\rm{s}}^2}} \right)} \\\\ = 6.859{\rm{ m/s}}\\\\ = 6.9{\rm{ m/s}}\\\end{array}$

Ans:

The velocity of the coaster is 6.9 m/s.