A small block of mass m=1.9kg slides, without friction, along the loop-the-loop track shown. The block starts from the point P a distance h=52.0m above the bottom of the loop of radius R=19.0m. What is the kinetic ennergy of the mass at the point A on the loop? What is the downward accelerations of the mass at point A? What is the minimum height h for which the block will reach point A on the loop without leaving the track?
To find the answers to these questions, let's break it down step by step.
1. Kinetic energy at point A:
The kinetic energy (KE) of an object is given by the formula KE = (1/2)mv^2, where m is the mass and v is the velocity. In this case, the block is sliding without friction, so its mechanical energy is conserved.
At point A, the block is at its lowest point on the loop. The mechanical energy of the block is equal to its gravitational potential energy at point P, which is given by mgh, where g is the acceleration due to gravity (9.8 m/s^2) and h is the height above the bottom of the loop.
Therefore, mgh = (1/2)mv^2.
Canceling out mass and solving for v^2, we get v^2 = 2gh.
Substituting the given values, v^2 = 2 * 9.8 * 52.
Therefore, the kinetic energy at point A is KE = (1/2)mv^2 = (1/2) * 1.9 * 2 * 9.8 * 52.
2. Downward acceleration at point A:
At point A, the block is moving in a circular path. The net centripetal force required to keep it in circular motion is provided by the gravitational force. Therefore, the downward acceleration at point A is equal to the acceleration due to gravity, which is 9.8 m/s^2.
3. Minimum height for reaching point A without leaving the track:
To determine the minimum height h required for the block to reach point A without leaving the track, we need to consider the minimum speed required at the top of the loop to maintain contact with the track.
At the top of the loop, the centripetal force is still provided by the gravitational force. Therefore, we have the equation: mgh = (1/2)mv^2.
Canceling out mass and solving for v^2, we get v^2 = 2gh.
To ensure the block maintains contact with the track, the speed at the top of the loop cannot be zero. So, the minimum speed is when v = 0.
Substituting v = 0, we get 0 = 2gh.
Therefore, h = 0. By setting h = 0, it means that the minimum height required for the block to reach point A without leaving the track is zero (i.e., the block has to start on or below the level of point A).
So, in summary:
1. The kinetic energy at point A is (1/2) * 1.9 * 2 * 9.8 * 52 joules.
2. The downward acceleration at point A is 9.8 m/s^2.
3. The minimum height required for the block to reach point A without leaving the track is zero.