A particle of mass m is initially situated at the point P inside a hemispherical surface of radius r . A horizontal acceleration of magnitude a₀ is suddenly produced on the particle in the horizontal direction. If gravitational accelertion is neglected, the time taken by particle to touch the sphere again is ?

To determine the time taken by the particle to touch the hemispherical surface again after being subjected to a horizontal acceleration, we can analyze the motion of the particle using basic principles of physics. Since we are neglecting gravitational acceleration, we can focus solely on the horizontal motion caused by the acceleration.

Understanding the Motion

When the particle experiences a horizontal acceleration \( a_0 \), it will move in a straight line in the direction of that acceleration. The particle starts from point P, which is located at a distance \( r \) (the radius of the hemisphere) from the center of the hemisphere. The key here is to find out how long it takes for the particle to travel horizontally until it makes contact with the hemispherical surface again.

Setting Up the Problem

Let’s break down the situation:

• The particle is initially at a point on the hemispherical surface.
• It begins to accelerate horizontally with an acceleration \( a_0 \).
• We need to find the time \( t \) it takes for the particle to reach the hemispherical surface again.

Using Kinematic Equations

In horizontal motion, we can use the kinematic equation that relates distance, initial velocity, acceleration, and time:

d = v_i t + (1/2) a t^2

Here, \( d \) is the horizontal distance the particle travels, \( v_i \) is the initial horizontal velocity (which is 0 since it starts from rest), \( a \) is the horizontal acceleration \( a_0 \), and \( t \) is the time we want to find.

Calculating the Distance

The distance \( d \) that the particle must travel to touch the hemispherical surface again can be determined by the geometry of the hemisphere. Since the particle starts at a point on the surface, the horizontal distance to the edge of the hemisphere is equal to the radius \( r \).

Thus, we can set \( d = r \). Plugging this into our kinematic equation gives:

r = 0 + (1/2) a_0 t^2

From this, we can simplify to:

r = (1/2) a_0 t^2

Solving for Time

Now, we can solve for \( t \):

t^2 = \frac{2r}{a_0}

Taking the square root of both sides, we find:

t = \sqrt{\frac{2r}{a_0}}

Final Result

The time taken by the particle to touch the hemispherical surface again, after being subjected to a horizontal acceleration \( a_0 \), is given by:

t = \sqrt{\frac{2r}{a_0}}

This formula shows that the time depends on 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 surface again.