To solve this problem, we need to analyze the motion of both the disc and the wedge. The scenario involves a disc projected upwards on a wedge, and we want to find the value of \( n \) in the given displacement equation. Let's break this down step by step.
Understanding the System
We have a wedge inclined at an angle \( \theta \) to the horizontal. The disc, which has the same mass as the wedge, is projected upwards with an initial velocity \( u \). As the disc moves up the wedge, it experiences gravitational forces and the normal force from the wedge's surface.
Forces Acting on the Disc
When the disc is on the wedge, the forces acting on it include:
- The gravitational force acting downwards, \( mg \).
- The normal force \( N \) exerted by the wedge, which acts perpendicular to the inclined surface.
The gravitational force can be resolved into two components relative to the wedge:
- Perpendicular to the wedge: \( mg \cos \theta \)
- Parallel to the wedge: \( mg \sin \theta \)
Equations of Motion
As the disc moves up the incline, it will decelerate due to the component of gravity acting down the slope. The net force acting on the disc along the incline is:
Net Force: \( F = -mg \sin \theta \)
Using Newton's second law, we can write:
Acceleration: \( a = -g \sin \theta \)
Finding the Maximum Height
To find the maximum height \( h \) that the disc reaches on the wedge, we can use the kinematic equation:
Final Velocity: \( v^2 = u^2 + 2a s \)
At the highest point, the final velocity \( v = 0 \), so we have:
0 = \( u^2 - 2g \sin \theta h \)
Rearranging gives us:
Height: \( h = \frac{u^2}{2g \sin \theta} \)
Displacement of the Wedge
As the disc moves up the wedge, the wedge itself will also move horizontally due to the conservation of momentum. The horizontal displacement \( x \) of the wedge can be related to the vertical height \( h \) reached by the disc. The relationship can be derived from the geometry of the system:
Horizontal Displacement: \( x = h \tan \theta \)
Substituting for \( h \) gives:
Displacement of Wedge: \( x = \frac{u^2 \tan \theta}{2g \sin \theta} \)
Final Equation and Finding \( n \)
The problem states that the displacement of the wedge when the disc is at the highest point is given by:
Given Displacement: \( x = \frac{u^2(1 + \sin^2 \theta)}{n g \sin 2 \theta} \)
We can equate our derived expression for \( x \) with the given expression:
\( \frac{u^2 \tan \theta}{2g \sin \theta} = \frac{u^2(1 + \sin^2 \theta)}{n g \sin 2 \theta} \)
By simplifying this equation, we can solve for \( n \). Noting that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \sin 2\theta = 2 \sin \theta \cos \theta \), we can substitute these into our equation:
After simplification, we find:
Value of \( n \): \( n = 2(1 + \sin^2 \theta) \cos \theta \)
Thus, the value of \( n \) is determined by the angle \( \theta \) and can be expressed as \( n = 2(1 + \sin^2 \theta) \cos \theta \). This result encapsulates the relationship between the motion of the disc and the displacement of the wedge in this intriguing physics problem.
