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A 250 kg motorcycle is driven around a 12 meter tall vertical circular track at a constant speed of 11 m/s.
Determine the normal and friction forces at the four points labeled in the diagram below.
Determine the minimum coefficient of static friction needed to complete the stunt as planned.
(The mass of the motorcycle includes the mass of the rider. Assume that aerodynamic drag and rolling resistance are negligible.)

The only forces present in this problem are weight (which always points down), normal (which always points toward the center of the loop), and friction (which is always tangential to the loop). The weight of the motorcycle never changes. The normal varies from a maximum at the bottom of the loop to a minimum at the top. As the motorcycle drives up the loop, the friction force acts along the direction of motion to keep gravity from slowing it down. As the motorcycle drives down the loop, the friction force acts opposite the direction of motion to keep gravity from speeding it up. The bike isn't speeding up or slowing down, but it is changing direction. This means the net force always points toward the center of motion.
Weight points down and normal points up, so the net force is their difference. The normal force points toward the center, so it should be given the positive value. The net force is the centripetal force.
∑F =  ma  
Ni − mg =  mv2  
Ni = m  ⎛
⎝ v2  + g ⎞

0 0 0
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