Final answer:
The difference between the normal forces exerted by the car on the passenger at the top and bottom of the vertical loop is 3.00mg.
Step-by-step explanation:
The normal force exerted by the car on a passenger varies at different points on a circular vertical loop-the-loop on a roller coaster. At the top of the loop, the normal force is greater than the passenger's weight because the centripetal acceleration is greater than the acceleration due to gravity. At the bottom of the loop, the normal force is less than the passenger's weight because the centripetal acceleration is less than the acceleration due to gravity.
To express the difference between the two normal forces, we can calculate the centripetal acceleration at the top and bottom of the loop using the given information. Let's assume the passenger's mass is represented by m, the acceleration due to gravity is represented by g, and the normal force at the top of the loop is represented by Ntop and at the bottom of the loop is represented by Nbottom
At the top of the loop, the centripetal acceleration is given by atop = g + 1.50g = 2.50g. The net force acting on the passenger at the top is Fnet,top = m * atop = m * 2.50g. This net force is equal to the sum of the normal force and the weight of the passenger, so we have Ntop + mg = m * 2.50g. Solving for Ntop, we get Ntop = 1.50mg.
At the bottom of the loop, the centripetal acceleration is given by abottom = g - 1.50g = -0.50g. The net force acting on the passenger at the bottom is Fnet,bottom = m * abottom = m * (-0.50g). This net force is equal to the sum of the normal force and the weight of the passenger, so we have Nbottom + mg = m * (-0.50g). Solving for Nbottom, we get Nbottom = -1.50mg.
The difference between the normal forces can be calculated by subtracting the normal force at the bottom from the normal force at the top:
Ntop - Nbottom = 1.50mg - (-1.50mg) = 1.50mg + 1.50mg = 3.00mg.