Question

A roller coaster car of mass 320 kg (including passengers) travels around a horizontal curve of radius \(35 \mathrm{m}\) Its speed is $16 \mathrm{m} / \mathrm{s} .$ What is the magnitude and direction of the total force exerted on the car by the track?

Short answer

The total force exerted on the roller coaster car by the track has a magnitude of 2339.2 N (rounded to a reasonable accuracy), and the direction of this force is towards the center of curvature of the coaster's circular path.

Step-by-step solution

Find the centripetal force

To find the centripetal force exerted on the car, we first need to find the centripetal acceleration of the car. Recall that the centripetal acceleration for an object moving in a circle of radius \(r\) at constant speed \(v\) can be expressed as \(a_c = \frac{v^2}{r}\). In this case, the car is moving at a speed of \(16\,\text{m/s}\) along a curved path with a radius of \(35\,\text{m}\). Plug these values into the formula to find the centripetal acceleration: \(a_c = \frac{(16\,\text{m/s})^2}{35\,\text{m}} = \frac{256\,\text{m}^2/\text{s}^2}{35\,\text{m}} = 7.31\,\text{m/s}^2\).

Apply Newton's second law of motion

We can now use Newton's second law of motion to calculate the resulting force exerted on the car by the track. This law states that the net force acting on an object is equal to the object's mass times its acceleration (\(F = ma\)). In this case, the net force acting on the car is the centripetal force we found earlier. The mass of the car is given as \(320\,\text{kg}\), and we calculated the centripetal acceleration to be \(7.31\,\text{m/s}^2\). Multiply these values to find the magnitude of the force: \(F = (320\,\text{kg})(7.31\,\text{m/s}^2) = 2339.2\,\text{N}\).

Determine the direction of the force

The direction of the centripetal force is always directed towards the center of the circle in which the object is moving. In this case, the total force exerted by the track on the car is directed towards the center of curvature of the circular path of the coaster.

Express the final answer

The magnitude of the total force exerted on the roller coaster car by the track is \(2339.2\,\text{N}\) (rounded to a reasonable accuracy), and the direction of this force is towards the center of curvature of the coaster's circular path.