a body is allowed to slide down a frictionless track from rest under gravity . The track end in a circular loop of diameter D . What should be the minimum height of the body in terms of D so that the body successfully complete the loop.
v^2/R = g or it loses contact
m g(h-D) = .5 mv^2
v^2 = g R = .5 g D
so
m g(h-D) = .5 m *.5 g D =.25 mgD
h-D = .25 D
h = 1.25 D
Energy is conserved so mgh =mgD
D is the maximum height
m and g are constant and the ony variable is h and h = D
all the PE at the start from height h is converted back to PE at height D
To determine the minimum height of the body required to successfully complete the loop, we need to consider the forces acting on the body at different positions on the track.
Let's assume that the body starts at the highest point of the loop (position A) with height h. At this point, the body has potential energy mg(h + r), where m is the mass of the body, g is the acceleration due to gravity, and r is the radius of the circular loop (D/2).
When the body reaches the lowest point of the track (position B) before entering the loop, it has converted all of its potential energy into kinetic energy. The kinetic energy at the bottom of the track is given by 1/2 mv², where v is the velocity of the body at that point.
For the body to successfully complete the loop, it must not lose contact with the track at any point. This means that the normal force acting on the body at the topmost point of the loop (position C) should not become zero. The minimum height needed for this condition can be derived using the centripetal force required to keep the body moving in a vertical circle.
At the topmost point of the loop, the force diagram for the body consists of the gravitational force (mg) acting downwards and the normal force (N) acting upwards. The centripetal force (Fc) is directed towards the center of the loop and is given by:
Fc = N - mg
To calculate the minimum height, we need to find the critical point where the normal force becomes zero. At this point, the centripetal force (Fc) must be equal to zero, as there is no normal force to counteract the gravitational force:
0 = N - mg
Solving this equation for N, we have:
N = mg
Substituting this back into the centripetal force equation:
Fc = mg - mg
As we know that Fc = mv²/r, where v is the velocity of the body, we have:
mv²/r = 0
This implies that v = 0, as the velocity of the body is zero at the topmost point of the loop. Therefore, the critical point at the topmost point of the loop is when the body is momentarily at rest.
Now, let's analyze the situation at the highest point (position A). At this point, the total energy of the body must be equal to the kinetic energy required to overcome gravity and reach the critical point (position C) without losing contact with the track.
The total energy can be expressed as:
mg(h + r) = 1/2 mv²
Since the velocity (v) is zero at position C, m cancels out and we have:
gh + gr = 0
Simplifying this equation, we find:
h = -r
Given that the radius (r) of the circular loop is equal to D/2, the minimum height required for the body to successfully complete the loop is:
h = -D/2
Therefore, the minimum height should be negative D/2 in terms of D. However, negative values for height are not physically meaningful. The actual minimum height required is D/2, which is equivalent to the radius of the loop.