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 is ................... (Ans. 1/root 2)

To solve the problem of comparing the times it takes for a particle to slide down two grooves arranged vertically along a disc, we need to analyze the motion of the particle in each groove. The grooves AB and CD are both vertical chords of the disc, and we can assume that the particle is sliding down under the influence of gravity. Let's break this down step by step.

Understanding the Setup

Imagine a disc with a radius R. The grooves AB and CD are vertical chords, meaning they are straight lines that connect two points on the circumference of the disc. The length of both grooves is the same, but their shapes and the angles at which they descend differ.

Analyzing the Motion

When the particle slides down the grooves, it experiences gravitational acceleration. The key to solving this problem lies in understanding how the angle of the groove affects the component of gravitational force acting along the groove.

• Groove AB: This groove is a straight vertical line. The gravitational force acting on the particle is directed straight down, and thus the entire force contributes to the acceleration along the groove.
• Groove CD: This groove, while also vertical, may have a different effective angle when considering the path taken by the particle. The gravitational force will still act downwards, but the path may not be as direct as in groove AB.

Calculating Time of Descent

To find the time taken for the particle to slide down each groove, we can use the principles of energy conservation or kinematics. However, a simpler approach is to consider the effective acceleration along each groove.

Using Energy Conservation

For both grooves, the potential energy lost by the particle as it descends is converted into kinetic energy. The potential energy at the top of the groove can be expressed as:

PE = mgh,

where m is the mass of the particle, g is the acceleration due to gravity, and h is the height from which the particle descends.

The kinetic energy at the bottom of the groove is:

KE = (1/2)mv².

Setting these equal gives us:

mgh = (1/2)mv².

From this, we can derive the velocity of the particle at the bottom of the groove:

v = √(2gh).

Time Calculation

The time taken to slide down each groove can be expressed in terms of the distance and velocity. For a straight groove like AB, the time Tab can be calculated as:

Tab = length of AB / v.

For groove CD, we need to consider the effective length and the angle, which may introduce a factor of √2 due to the geometry of the path. Thus, we can express Tcd as:

Tcd = length of CD / (v/√2).

Finding the Ratio

Now, we can find the ratio of the times:

Tab/Tcd = (length of AB / v) / (length of CD / (v/√2)) = (length of AB * √2) / length of CD.

Since the lengths of AB and CD are equal, we can simplify this to:

Tab/Tcd = √2.

However, since we are looking for the inverse ratio, we find:

Tab/Tcd = 1/√2.

Final Result

Thus, the ratio of the time taken for the particle to slide down the grooves is:

Tab/Tcd = 1/√2.

This result shows how the geometry of the grooves affects the time of descent, illustrating the fascinating interplay between physics and geometry in motion problems.