A 3.9 g mass is released from rest at C which has a height of 0.9 m above the base of a loop-the-loop and a radius of 0.4 m.
The acceleration of gravity is 9.8 m/s^2. Find the normal force pressing on the track at A, where A is at the same level as the center
of the loop.
Answer in units of N.
b. Consider a different situation when the initial height atC has not yet been specified. What is the minimum kinetic energy of the block at B, which is located at the top of the
loop, so that the block can pass by this point without falling off from the track?
Answer in units of J.
To find the normal force pressing on the track at point A, we need to consider the forces acting on the block at that point. The only forces involved are the gravitational force (mg) and the normal force (N). At point A, the normal force and the weight of the block have the same magnitude but act in opposite directions.
First, let's calculate the weight of the block:
Weight = mass × acceleration due to gravity
Weight = 3.9 g × 9.8 m/s² = 38.22 N
Since the normal force is equal in magnitude but opposite in direction to the weight of the block, the normal force can be found as 38.22 N.
Now let's move on to the second part of the question. To determine the minimum kinetic energy required for the block to pass point B without falling off, we need to consider energy conservation.
At the highest point of the loop (point B), the block will have gravitational potential energy (mgh) and kinetic energy (1/2 mv²). For the block to remain on the track, the total mechanical energy at point B must be equal to or greater than the gravitational potential energy at point C.
First, let's calculate the gravitational potential energy at point C:
Gravitational Potential Energy at C = mass × gravitational acceleration × height
Gravitational Potential Energy at C = 3.9 g × 9.8 m/s² × 0.9 m = 33.033 J
The minimum kinetic energy required at point B can be calculated by subtracting the gravitational potential energy at point C from the total mechanical energy at point B:
Minimum Kinetic Energy at B = Total Mechanical Energy at B - Gravitational Potential Energy at C
Since the block is at rest at point C, its total mechanical energy at B is equal to its gravitational potential energy at C:
Minimum Kinetic Energy at B = Gravitational Potential Energy at C - Gravitational Potential Energy at C
Minimum Kinetic Energy at B = 0 J
Therefore, the minimum kinetic energy required at point B is 0 J. This means that as long as the block has any amount of kinetic energy (even the slightest), it will be able to pass point B without falling off the track.