Question

A roller coaster has a vertical loop with radius \(29.5 \mathrm{m}\) With what minimum speed should the roller coaster car be moving at the top of the loop so that the passengers do not lose contact with the seats?

Short answer

Answer: The minimum speed the roller coaster car should be moving at the top of the loop is approximately 17 m/s.

Step-by-step solution

Understanding the Forces Involved

At the top of the loop, the centripetal force acts towards the center of the loop, and the force of gravity acts downward. To find the minimum speed to avoid losing contact, we need to set the centripetal force equal to gravitational force as that will be the point where the passengers are on the verge of losing contact.

Write the Centripetal Force Equation

The centripetal force can be calculated using the formula: \(F_c = m \frac{v^2}{r}\) where \(F_c\) is the centripetal force, \(m\) is the mass, \(v\) is the velocity, and \(r\) is the radius of the circular path.

Write the Gravitational Force Equation

The gravitational force can be calculated using the formula: \(F_g = mg\) where \(F_g\) is the gravitational force, \(m\) is the mass, and \(g\) is the acceleration due to gravity (approximately \(9.8\,\mathrm{m/s^2}\)).

Set Centripetal Force Equal to Gravitational Force

To find the minimum speed that the roller coaster car should be moving at the top of the loop, we need to set \(F_c\) equal to \(F_g\): \(m \frac{v^2}{r} = mg\)

Solve for Minimum Velocity

We can solve the previous equation for the minimum velocity by dividing both sides by \(m\) and multiplying by \(r\): \(v^2 = rg\) Now, we can find the minimum velocity \(v\) by taking the square root of both sides: \(v = \sqrt{rg}\)

Calculate the Minimum Velocity

Now we can plug in the values for \(r\) and \(g\): \(v = \sqrt{(29.5\,\mathrm{m})(9.8\,\mathrm{m/s^2})}\) Calculating the result: \(v \approx 17\,\mathrm{m/s}\) The minimum speed the roller coaster car should be moving at the top of the loop to prevent the passengers from losing contact with the seats is approximately \(17\,\mathrm{m/s}\).