To determine the time taken by a particle of mass m, initially at point P inside a hemispherical surface of radius r, to touch the sphere again after being subjected to a horizontal acceleration of magnitude a0, we can analyze the motion of the particle using basic principles of physics.
Understanding the Motion
When the particle is given a horizontal acceleration, it will start moving in the horizontal direction while simultaneously being influenced by the geometry of the hemisphere. Since we are neglecting gravitational acceleration, the only force acting on the particle is the horizontal acceleration a0.
Breaking Down the Motion
Initially, the particle is at rest at point P. Once the horizontal acceleration is applied, the particle will begin to move horizontally. The key points to consider are:
- The particle moves horizontally with constant acceleration a0.
- The distance it needs to travel horizontally before it touches the hemisphere again is determined by the radius r of the hemisphere.
Calculating the Time
To find the time taken for the particle to reach the hemisphere again, we can use the equations of motion. The horizontal displacement (x) of the particle can be described by the equation:
x = 0.5 * a0 * t²
Since the particle starts from point P, which is at a distance r from the edge of the hemisphere, we set x equal to r:
r = 0.5 * a0 * t²
Rearranging this equation to solve for time (t), we get:
t² = (2r) / a0
Taking the square root of both sides gives us:
t = √(2r / a0)
Final Result
Thus, the time taken by the particle to touch the hemisphere again after the horizontal acceleration is applied is:
t = √(2r / a0)
This result shows how the time depends on both the radius of the hemisphere and the magnitude of the horizontal acceleration. The larger the radius or the smaller the acceleration, the longer it will take for the particle to make contact with the hemisphere again.
