To solve this problem, we need to analyze the forces acting on the particle as it moves in a vertical circle while being attached to a hinged rod. The key here is to understand the conditions under which the net force on the particle is horizontal and how the angle of inclination of the rod affects this scenario.
Understanding the Forces at Play
When the particle is moving in a vertical circle, it experiences two main forces: gravitational force acting downward and the tension in the rod acting along the rod. The net force on the particle can be expressed as the vector sum of these forces.
Setting Up the Problem
Let’s denote the following:
- m: mass of the particle
- g: acceleration due to gravity
- T: tension in the rod
- θ: angle of inclination of the rod with the horizontal, which is given as sin-1(1/x)
At the point where the net force is horizontal, the vertical component of the forces must balance out, while the horizontal components must add up to provide the necessary centripetal force for circular motion.
Force Components
We can break down the forces into their components:
- The gravitational force acting downward is Fg = mg.
- The tension can be resolved into two components:
- Vertical component: Ty = T cos(θ)
- Horizontal component: Tx = T sin(θ)
Condition for Horizontal Net Force
For the net force to be horizontal, the vertical forces must balance out:
T cos(θ) = mg
And the horizontal force must provide the centripetal force required for circular motion:
T sin(θ) = m(v2/r)
Relating the Forces
From the first equation, we can express tension in terms of gravitational force:
T = mg / cos(θ)
Substituting this expression for tension into the second equation gives:
(mg / cos(θ)) sin(θ) = m(v2/r)
We can simplify this to:
g tan(θ) = v2/r
Using the Angle of Inclination
Now, substituting θ = sin-1(1/x) into the equation:
- tan(θ) = sin(θ) / cos(θ) = (1/x) / sqrt(1 - (1/x)2)
Thus, we have:
g * (1/x) / sqrt(1 - (1/x)2) = v2/r
Finding the Value of x
To find the value of x, we need to analyze the condition when the net force is purely horizontal. This occurs when the angle θ leads to a specific relationship between the gravitational force and the centripetal force. By manipulating the equations, we can derive:
x = sqrt(1 + (v2 / g * r)2)
Given that the problem states the net force is horizontal, we can deduce that for the particle to complete the vertical revolution, the value of x must satisfy the relationship derived from the forces acting on the particle.
Final Calculation
To find the exact numerical value of x, we would typically need additional information about the velocity v and the radius r of the circular path. However, if we assume that the particle is just at the threshold of completing the vertical circle, we can set the centripetal force equal to the gravitational force at the top of the circle, leading to:
x = 2
Thus, under the conditions provided, the value of x is 2, indicating that the angle of inclination of the rod is such that it allows the particle to maintain a horizontal net force while completing its vertical revolution.
