A disc arranged in a vertical plane has two groves of same length directed along the vertical chord AB and CD as shown. The same particle slides down along AB and CD. The ratio of time Tab/Tcd

To determine the ratio of the times taken for a particle to slide down two grooves, AB and CD, on a vertical disc, we need to analyze the motion of the particle in both grooves. The grooves are oriented along the vertical chord of the disc, and since they are of the same length, we can focus on the forces acting on the particle and the geometry of the grooves.

Understanding the Motion

When a particle slides down a groove under the influence of gravity, its motion can be described using principles of physics, particularly energy conservation and kinematics. The key factors affecting the time taken to slide down each groove are the angle of inclination and the path length.

Analyzing the Grooves

• Groove AB: This groove is inclined at a certain angle with respect to the vertical. The component of gravitational force acting along the groove will determine the acceleration of the particle.
• Groove CD: Similarly, this groove has its own angle of inclination, which will also affect the acceleration of the particle as it slides down.

Acceleration Calculation

The acceleration of the particle sliding down each groove can be expressed as:

• For groove AB: a_{AB} = g \sin(\theta_{AB})
• For groove CD: a_{CD} = g \sin(\theta_{CD})

Here, g is the acceleration due to gravity, and θ represents the angle of inclination of each groove with respect to the vertical.

Time of Descent

The time taken to slide down a distance s under constant acceleration can be derived from the kinematic equation:

s = ut + (1/2)at²

Assuming the initial velocity u is zero (the particle starts from rest), we can simplify this to:

s = (1/2)at²

Rearranging gives us:

t = √(2s/a)

Substituting the accelerations for each groove, we find:

• T_{AB} = √(2s/a_{AB}) = √(2s/(g \sin(\theta_{AB})))
• T_{CD} = √(2s/a_{CD}) = √(2s/(g \sin(\theta_{CD})))

Finding the Ratio

Now, we can find the ratio of the times:

Tab/Tcd = √(g \sin(θ_{CD})/g \sin(θ_{AB})) = √(sin(θ_{CD})/sin(θ_{AB}))

This ratio indicates how the angles of inclination affect the time taken for the particle to slide down each groove. If both grooves are symmetrical and have the same length, the ratio will depend solely on the sine of the angles of inclination.

Conclusion

In summary, the ratio of the times taken for the particle to slide down the grooves AB and CD can be expressed as:

Tab/Tcd = √(sin(θ_{CD})/sin(θ_{AB}))

This relationship highlights the importance of the angle of inclination in determining the dynamics of the sliding motion. By understanding these principles, we can predict the behavior of particles in various physical scenarios.