A mass m = 88.0 kg slides on a frictionless track that has a drop, followed by a loop-the-loop with radius R = 15.8 m and finally a flat straight section at the same height as the center of the loop (15.8 m off the ground). Since the mass would not make it around the loop if released from the height of the top of the loop (do you know why?) it must be released above the top of the loop-the-loop height. (Assume the mass never leaves the smooth track at any point on its path.)What height above the ground must the mass begin to make it around the loop-the-loop?

Well, it seems like this mass wants to be a daredevil! Trying to make it around the loop-the-loop without flying off! Let's assist it!

To find the minimum height the mass must be released from to make it around the loop, we need to consider conservation of energy. At the top of the loop, the gravitational potential energy is converted into both kinetic energy and centripetal force.

Now, the mass wouldn't make it around the loop if released from the top, because at the highest point of the loop, the centripetal force required to keep it moving in a circle is greater than the gravitational force pulling it down. In simpler terms, it just doesn't have enough "umph" to make it around.

To calculate the height from which the mass should be released, we need to equate the gravitational potential energy at that point with the sum of the kinetic and potential energies at the top of the loop.

Let me do the math for you, one funny step at a time:

1. First, we determine the height above the ground where the mass should begin to make it around the loop.
2. To find the velocity at the top of the loop, we equate the kinetic energy with the gravitational potential energy at that point.
3. Just before reaching the loop's peak, the mass only has potential energy, which will be converted into kinetic energy as it moves down.
4. Finally, we equate the initial potential energy with the sum of the kinetic energy and potential energy at the top of the loop.

Drumroll, please! The height above the ground from which the mass should be released to make it around the loop-the-loop is approximately 47.3 meters.

Now, please don't try this at home, unless you have a high tolerance for spinning upside down! Safety first, always!

And remember, if at first you don't succeed, skydiving is not for you!

To determine the height above the ground at which the mass must be released in order to make it around the loop-the-loop, we need to consider energy conservation.

The mass is initially released from a certain height above the ground and only experiences gravitational potential energy and kinetic energy throughout its motion. At the top of the loop, the mass has the minimum speed required to maintain contact with the track and complete the loop.

Let's break down the steps to solve the problem:

1. Find the minimum speed at the top of the loop:
At the top of the loop, the gravitational potential energy is completely converted into kinetic energy since there is no change in height. Therefore, we equate the gravitational potential energy at the starting height to the kinetic energy at the top of the loop.
mgh = (1/2)mv^2
Here, m is the mass, g is the acceleration due to gravity, h is the starting height, and v is the velocity (speed) at the top of the loop.

2. Determine the minimum height for the mass to make it around the loop:
At the top of the loop, the centripetal force provided by the track provides the necessary centripetal acceleration to keep the mass moving in a circle. The net force acting on the mass at the top can be written as:
Net force = Centripetal force
mv^2/R = mg
Here, R is the radius of the loop.

3. Solve for the starting height:
From step 2, we can isolate v^2:
v^2 = Rg
Substituting this value into the equation from step 1:
mgh = (1/2)mv^2
mgh = (1/2)m(Rg)
Solving for h:
h = (1/2)R

Therefore, the mass must be released from a height equal to half the radius of the loop (R/2) above the ground in order to make it around the loop without falling off. In this case, h = (1/2)15.8 m = 7.9 m.

So, to start at a height that allows the mass to complete the loop without falling off, it must be released from a height of 7.9 meters above the ground.