Question

The marble rolls down the track and around a loop-the-loop of radius R. The marble has mass m and radius r. What minimum height h must the track have for the marble to make it around the loop-the-loop without falling off?

Answer 1

The minimum height the track must have for the marble to make it around the loop-the-loop without falling off is 3R/2, where R is the radius of the loop.

Step-by-step explanation:

In order for the marble to make it around the loop-the-loop without falling off, the track must have a minimum height h.

To determine this, we can consider the forces acting on the marble. At the top of the loop, the centrifugal force must be greater than or equal to the gravitational force to keep the marble moving in a circle. The centrifugal force is given by:

mv^2/R

where m is the mass of the marble, v is its velocity, and R is the radius of the loop. The gravitational force is given by:

mg

where m is the mass of the marble and g is the acceleration due to gravity.

Thus, we can set these two forces equal to each other:

mv^2/R = mg

Canceling out the mass of the marble, we get:

v^2/R = g

Solving for v, we get:

v = sqrt(Rg)

Now, at the top of the loop, the total height, h, is equal to the sum of the radius of the loop and the height from the bottom of the loop to the top. We can write this as:

h = R + height from bottom to top

The height from the bottom to the top can be obtained using the conservation of energy. At the bottom of the loop, the marble has only kinetic energy, and at the top of the loop, it has both kinetic and potential energy. Since there is no energy lost due to friction, we can equate the initial kinetic energy to the final total mechanical energy:

1/2mv^2 = mg(h - R)

Simplifying this equation, we get:

v^2 = 2g(h - R)

Substituting the expression we derived for v, we get:

Rg = 2g(h - R)

Cancelling out g, we get:

R = 2(h - R)

Simplifying this equation, we get:

R = 2h - 2R

Combining like terms, we get:

3R = 2h

Solving for h, we get:

h = 3R/2

Therefore, the minimum height the track must-have for the marble to make it around the loop-the-loop without falling off is 3R/2.

Answer 2

The minimum height h that the track must have for the marble to make it around the loop-the-loop without falling off is 1/2 times the square of the velocity at the bottom of the loop divided by the acceleration due to gravity.

Step-by-step explanation:

A marble will make it around a loop-the-loop without falling off if the track has a minimum height of h. To determine the minimum height required, we need to consider the energy conservation principles and the forces acting on the marble. At the maximum height of the loop, the marble will experience a normal force from the track and its weight acting downwards. At the minimum height, the normal force becomes zero, which means the marble is barely on the track.

Let's assume the marble starts at a height h and its velocity at the bottom of the loop is v. At the maximum height of the loop, the sum of the marble's gravitational potential energy and kinetic energy equals its total mechanical energy. At the minimum height, only the marble's kinetic energy contributes to the total mechanical energy.

To find the minimum height of the track, we can equate the marble's potential energy at the maximum height to its kinetic energy at the minimum height:

mgh = ½mv²

Where m is the mass of the marble, g is the acceleration due to gravity, and v is the velocity of the marble at the bottom of the loop. Rearranging the equation, we get:

h = ½v²/g

Therefore, the minimum height h that the track must have for the marble to make it around the loop-the-loop without falling off is ½v²/g.