To solve the problem of a particle sliding down a smooth inclined plane while the elevator is accelerating upwards, we need to analyze the forces acting on the particle and apply the equations of motion. Let's break it down step by step.

Understanding the Setup

We have an inclined plane with an angle of elevation θ and a length L. The elevator is moving upwards with an acceleration a. Since we are considering an inertial frame (the observer on the ground), we need to account for the effective gravitational force acting on the particle due to the upward acceleration of the elevator.

Effective Gravitational Force

In the inertial frame, the effective gravitational acceleration acting on the particle can be expressed as:

• The actual gravitational acceleration, g (approximately 9.81 m/s²).
• The upward acceleration of the elevator, a.

Thus, the effective gravitational acceleration acting on the particle becomes:

g_eff = g + a

Components of Forces on the Incline

When the particle is on the incline, the effective gravitational force can be resolved into two components:

• Perpendicular to the incline: g_eff * cos(θ)
• Parallel to the incline: g_eff * sin(θ)

Only the parallel component contributes to the acceleration of the particle down the incline.

Acceleration of the Particle

The acceleration of the particle down the incline, denoted as a_particle, is given by the parallel component of the effective gravitational force:

a_particle = g_eff * sin(θ) = (g + a) * sin(θ)

Applying Equations of Motion

Now that we have the acceleration, we can use the equations of motion to find the time taken by the particle to slide down the incline. The relevant equation for uniformly accelerated motion is:

s = ut + (1/2)at²

Where:

• s = distance along the incline (L)
• u = initial velocity (0, since the particle starts from rest)
• a = acceleration (a_particle)
• t = time taken

Substituting the values, we have:

L = 0 * t + (1/2) * (g + a) * sin(θ) * t²

This simplifies to:

L = (1/2) * (g + a) * sin(θ) * t²

Solving for Time

To find the time t, we can rearrange the equation:

t² = (2L) / ((g + a) * sin(θ))

Taking the square root gives us:

t = √((2L) / ((g + a) * sin(θ)))

Final Expression

Thus, the time taken by the particle to reach the bottom of the incline is:

t = √((2L) / ((g + a) * sin(θ)))

This formula allows you to calculate the time based on the length of the incline, the angle of elevation, the gravitational acceleration, and the upward acceleration of the elevator. By plugging in the appropriate values, you can find the time taken for the particle to slide down the incline.