Question

Suppose you are riding on a roller coaster, which moves through a vertical circular loop. Show that your apparent weight at the bottom of the loop is six times your weight when you experience weightlessness at the top, independent of the size of the loop. Assume that friction is negligible.

Short answer

Question: Show that the apparent weight of a rider at the bottom of a circular vertical loop on a roller coaster is six times their weight when they experience weightlessness at the top, regardless of the loop's size. Answer: The apparent weight at the bottom of the loop is twice the weight when the rider experiences weightlessness at the top, not six times. This ratio is independent of the loop's size and is equal to 2, not 6.

Step-by-step solution

Circular motion and centripetal force

Since the roller coaster is moving in a circular loop, its motion is a combination of a vertical and a horizontal movement. It experiences a centripetal force towards the center of the loop, which must balance the gravitational force when an object is in weightlessness at the top of the loop. The centripetal force F_c acting on a body of mass m and moving with a speed v in a circle of radius r can be expressed as: \[F_c = mr\omega^2\] where \(\omega\) is the angular velocity of the body.

Considering the forces at the bottom of the loop

At the bottom of the loop, two forces act on the rider: the gravitational force acting downward and the upward centripetal force provided by the rails, which together make up the apparent weight. Summing the forces acting on the rider, we get: \[F_{app,bottom} = F_c + mg\] where \(F_{app,bottom}\) is the apparent weight at the bottom of the loop and mg is the gravitational force.

Considering the forces at the top of the loop

At the top of the loop, the rider experiences weightlessness, which means the downward gravitational force must be equal to the upward centripetal force: \[F_c = mg\]

Substituting centripetal force in step 2

From step 3, we get the value of centripetal force \(F_c = mg\), substituting it in the expression obtained in step 2, we get: \[F_{app,bottom} = mg + mg\] \[F_{app,bottom} = 2mg\]

Calculating the ratio of apparent weight at the bottom and top

We found that the apparent weight at the bottom is \(2mg\), and at the top, the rider experiences weightlessness (\(mg\)). Now we can calculate the ratio of the apparent weight at the bottom(\(F_{app,bottom}\)) to the apparent weight at the top(\(F_{app,top}\)): \[\frac{F_{app,bottom}}{F_{app,top}} = \frac{2mg}{mg}\]

Final result

Simplifying the expression from Step 5, we get: \[\frac{F_{app,bottom}}{F_{app,top}} = 2\] The apparent weight at the bottom of the loop is twice the weight when the rider experiences weightlessness at the top. This contradicts the claim that it is six times, showing that the original statement in the exercise is incorrect. The ratio is indeed independent of the loop's size and is equal to 2, not 6.