Exercise 1: a) What is the apparent weight for a roller coaster rider at the bottom of a loop vs at the top of a loop? b) What is the min speed the car must pass along the loop so that the rider does not fall out while upside down at the top of the loop?
how big is the loop?
Only the max radius given R from top to center (loop not circular)
m v^2/R = m g to just barely not fall out at top
top: mg down and m v^2/R up
bottom: mg down and mv^2/R down
To answer part (a) of the question, we need to understand the concept of apparent weight. Apparent weight is the force experienced by an object or a person due to the contact forces acting on it. In the case of a roller coaster rider, it refers to the force they feel pushing them against their seat.
When a roller coaster moves through a loop, the rider experiences different apparent weights at the top and the bottom of the loop. At the top of the loop, the rider is upside down, with their head pointing towards the center of the loop, while at the bottom of the loop, the rider is right-side up.
At the bottom of the loop:
The apparent weight will be greater than the actual weight of the rider. This is because in addition to the force of gravity pulling the rider downwards, there is an additional force pushing them upwards called the centrifugal force. This force is caused by the circular motion of the roller coaster and acts towards the center of the loop, helping to keep the rider on their seat.
At the top of the loop:
The apparent weight will be less than the actual weight of the rider. This is because the centrifugal force is now acting outwards away from the center of the loop. While gravity is still pulling the rider downwards, the centrifugal force is partially canceling out the effect of gravity. As a result, the rider will feel lighter or even experience a sensation of weightlessness.
To answer part (b) of the question, we need to determine the minimum speed required for the rider to remain in contact with their seat without falling out at the top of the loop. This speed can be calculated using the concept of centripetal force.
Centripetal force is the force required to keep an object moving in a circular path. In the case of a roller coaster loop, the centripetal force at the top of the loop is provided by the normal force exerted by the seat on the rider.
To prevent the rider from falling out, the centripetal force at the top of the loop must be equal to or greater than the gravitational force acting on the rider.
Equating the centripetal force to the gravitational force, we have:
Normal Force = Gravitational Force
m * v^2 / r = m * g
Where:
m is the mass of the rider,
v is the speed of the roller coaster at the top of the loop,
r is the radius of the loop, and
g is the acceleration due to gravity.
Rearranging the equation, we get:
v^2 = r * g
Taking the square root of both sides, we find:
v = √(r * g)
So, the minimum speed required for the rider to not fall out at the top of the loop is the square root of the product of the radius of the loop and the acceleration due to gravity.