How do I solve this?
For the normal force in the drawing to have the same magnitude at all points on the vertical track, the stunt driver must adjust the speed to be different at different points. Suppose, for example, that the track has a radius of 2.87 m and that the driver goes past point 1 at the bottom with a speed of 22.4 m/s. What speed must she have at point 3, so that the normal force at the top has the same magnitude as it did at the bottom?
at the top, normal force=mg-v^2/r
at the bottom, normalforce=mg+v^2/r
To solve this problem, we need to use the concept of centripetal force. The normal force at the bottom of the track is equal to the weight of the stunt driver plus the centripetal force necessary to keep her in circular motion. The normal force at the top of the track is equal to the weight of the stunt driver minus the centripetal force. Since the radius at the bottom and top of the track are the same, the centripetal force is the same at both points.
Let's break down the problem into steps:
Step 1: Find the weight of the stunt driver.
The weight of an object is given by the formula:
weight = mass * gravitational acceleration
Since we are not given the mass of the stunt driver, we cannot calculate the weight directly. However, we can assume a mass for the driver to continue with the problem. Let's assume the mass of the stunt driver is 70 kg. The gravitational acceleration is approximately 9.8 m/s^2. Now, calculate the weight using the formula.
weight = 70 kg * 9.8 m/s^2
Step 2: Calculate the centripetal force at the bottom of the track.
The centripetal force required to keep an object moving in a circular path is given by the formula:
centripetal force = mass * (velocity^2 / radius)
In this case, we know the radius of the track at the bottom is 2.87 m and the velocity is 22.4 m/s. Assuming the mass of the stunt driver is still 70 kg, let's calculate the centripetal force.
centripetal force = 70 kg * (22.4 m/s)^2 / 2.87 m
Step 3: Calculate the normal force at the bottom of the track.
The normal force at the bottom of the track is equal to the weight plus the centripetal force.
normal force at the bottom = weight + centripetal force
Step 4: Calculate the normal force at the top of the track.
The normal force at the top of the track is equal to the weight minus the centripetal force.
normal force at the top = weight - centripetal force
Step 5: Calculate the centripetal force at the top of the track.
Since the centripetal force is the same at the bottom and top of the track, we can use the centripetal force calculated at the bottom of the track to find the speed at the top. Rearranging the centripetal force formula, we get:
centripetal force = mass * (velocity^2 / radius)
Rearranging again to solve for the velocity, we have:
velocity = sqrt(centripetal force * radius / mass)
Substituting the known values, we have:
velocity at the top = sqrt(centripetal force * 2.87 m / 70 kg)
Now, calculate the velocity at the top of the track. This will be the speed that the stunt driver must have at point 3 to ensure that the normal force at the top has the same magnitude as it did at the bottom.