Shadow Oscillation

The problem describes a ball moving with constant speed in a circular path in a vertical plane. The system is illuminated from the top, casting a shadow of the ball on the ground. We need to determine where the probability of finding this shadow is maximum.

When an object moves in a circle and is illuminated by parallel light rays perpendicular to the plane of motion, its shadow on a screen perpendicular to the light rays executes Simple Harmonic Motion (SHM).

In this case, the ball moves in a vertical circular path, and the illumination is from the top. This means the light rays are essentially vertical, perpendicular to the horizontal ground. The shadow on the ground is the projection of the ball's position onto the ground. As the ball moves around the vertical circle, its horizontal position (on the ground) will oscillate back and forth along a straight line segment. This oscillatory motion of the shadow is Simple Harmonic Motion.

Understanding Probability in SHM

In Simple Harmonic Motion, the speed of the oscillating object is not constant. The speed is maximum at the mean position (center of oscillation) and minimum (zero) at the extreme positions (the points where the object reverses direction).

The probability of finding the object at a particular point is related to how long the object spends in the vicinity of that point. The longer the object spends near a point, the higher the probability of finding it there.

• At the mean point, the speed is maximum, so the object passes through this point quickly. The time spent near the mean point is minimum.
• At the extreme points, the speed is minimum (zero), so the object momentarily stops and reverses direction. The object spends the maximum amount of time near the extreme points as it slows down before turning around.

Since the shadow, undergoing SHM, spends the most time near the extreme points of its oscillation, the probability of finding the shadow is maximum at these extreme points.

Conclusion

Based on the properties of Simple Harmonic Motion, the shadow spends the most time at its turning points, which are the extreme points of the oscillation. Therefore, the probability of finding the shadow is highest at the extreme points.