1. A rollercoaster cart with a mass of 18 kg is travelling around a vertical loo

Question

1. A rollercoaster cart with a mass of 18 kg is travelling around a vertical loop (circle) with a radius of 9 meters. At the top, it's traveling at the minimum speed needed so that it remains in contact with the track. If at some short time later it is at the side of the loop (halfway between the top and bottom), what is the normal force on the cart, in Newtons, if it loses 143 Joules of work to non-conservative forces (friction) as it traveled to the side of the loop?

2. A rubber ball with a mass of 0.63 kg is released from rest from a height of 5 meters above the ground. Due to the interaction with the ground, each time it bounces it loses some of its total mechanical energy to non-conservative forces and it bounces to 71 percent of its height before it hit the ground. After 4 bounces, how much total energy is lost, in Joules?

3.An object with a mass of 6 kg is thrown straight upward on Earth at a speed of 43.74 m/s. Due to air resistance, i.e., non-conservative work, it only reaches a maximum displacement of 38.4 meters. If the object were thrown at the same speed straight upwards on another planet where the mass of the planet was a factor of 2.9 larger than that of earth and its radius was a factor of 3.1 smaller than earth, what would its maximum vertical displacement be, in meters, on this new planet if the amount of energy lost to non-conservative forces was a factor of 1.9 smaller than that of earth?

4.An object with a mass of 10 kg starts from rest on a horizontal surface with friction and the coefficient of kinetic friction is 0.53. Two additional forces are applied to it. One force of 19 Newtons is applied straight downward and another force of 76 Newtons is applied in the +x direction, parallel to the surface. This force parallel to the surface is enough to overcome static friction and move the mass. After the mass has traveled 9 meters over this horizontal surface (the entire surface has friction), these two additional forces are not present any more and it encounters a frictionless quarter-circular ramp. It reaches the top of ramp ("side" of the circle) before it stops. What is the radius, in meters, of the quarter-circular ramp?

5.Similar to the roller coaster problem discussed in class, suppose a roller coaster starts from rest at the top of an incline which is inclined at 41 degrees, but this time the entire surface of the incline has friction with a coefficient of kinetic friction of 0.51 (the angle of the incline is great enough such that the cart overcomes static friction). After the incline, it encounters a horizontal frictionless surface, and then a vertical loop with a radius of 8 meters. (Exactly like the example in class except the incline has friction.) If the cart is to barely make it around the top of the loop, what does the minimum length (not height) of the incline need to be in meters?

6. A mass of 10 kg is compressed into a spring on a horizontal surface. It is compressed into the spring 0.86 meters and held still. The force required to do this is 675 Newtons. It is then released from rest and the mass travels over a originally frictionless, horizontal surface, but at the end of the surface there is some friction with a coefficient of kinetic friction of 0.72. It then travels up a frictionless incline with an inclination angle of 16 degrees. It travels a distance of 4 meters up the incline before it stops. What was the length, in meters, of the portion of the horizontal surface that had friction?

Explanation / Answer

1)

m = 18 Kg

r = 9 m

at the top of the loop

velocity is v , for minimum speed

as m * v^2/r = m * g

v^2 = g * r

NOw, let the velocity when it reaches the side is vf

Using work energy theorum

0.50 * 18 * (vf^2 - v^2) = 18 * 9.8 * r - 143

0.50 * 18 * (vf^2 - (9.8 * 9)) = 18 * 9.8 * 9 - 143

solving for vf

vf = 15.7 m/s

for the normal force at the location

normal force = m * v^2/r

normal force = 18 * 15.7^2/9.8

normal force = 452 N

the normal force acting on the side is 452 N

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