Two bodies of masses m1 = 5 kg and m2 = 3 kg are connected by a light string going over a smooth light pulley on a smooth inclined plane as shown in the figure. The system is at rest. The force exerted by the inclined plane on the body of mass m1 will be
[Take g = 10 ms–2]
Two bodies of masses m1 = 5 kg and m2 = 3 kg are connected by a light string going over a smooth light pulley on a smooth inclined plane as shown in the figure. The system is at rest. The force exerted by the inclined plane on the body of mass m1 will be
[Take g = 10 ms–2]
- 30 N
- 40 N
- 50 N
- 60 N
The Correct Option is B
Approach Solution - 1
To find the force exerted by the inclined plane on the body of mass \(m_1\), we need to calculate the normal force since the system is at rest. First, let's identify the forces acting on \(m_1\):
- Gravitational force (\(m_1g\)) acting vertically downward.
- Normal force (\(N\)) acting perpendicular to the inclined plane.
Since the system is at rest, the net force along the plane and perpendicular to the plane must be zero. Therefore, we need to resolve the gravitational force into two components:
- Parallel to the plane: \(m_1g \sin \theta\)
- Perpendicular to the plane: \(m_1g \cos \theta\)
Because the system is at rest, the parallel component of the gravitational force is balanced by tension in the string, and the normal force balances the perpendicular component of the gravitational force.
Thus, the normal force \(N\) is given by:
\(N = m_1g \cos \theta\)
Substitute the given values \(m_1 = 5 \, \text{kg}\) and \(g = 10 \, \text{m/s}^2\). However, we still need the angle \(\theta\).
Since no angle \(\theta\) is given directly, and it's a typical balanced problem where \(m_1g \sin \theta = m_2g\) for balance (at rest), we can infer:
\(m_1g \cos \theta = 40 \, \text{N}\)
Hence the correct answer is 40 N.
Approach Solution -2
m2g = m1gsinθ …(i)
N = m1gcosθ …(ii)
⇒\(\frac{N}{m_2g}=cotθ\)
⇒N=3×10×cotθ
⇒N=3×10×\(\frac{4}{3} \)(∵sinθ=\(\frac{3}{5} \))
⇒ N = 40 N
So, the correct option is (B): 40 N.
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Concepts Used:
Center of Mass
The center of mass of a body or system of a particle is defined as a point where the whole of the mass of the body or all the masses of a set of particles appeared to be concentrated.
The formula for the Centre of Mass:
Center of Gravity
The imaginary point through which on an object or a system, the force of Gravity is acted upon is known as the Centre of Gravity of that system. Usually, it is assumed while doing mechanical problems that the gravitational field is uniform which means that the Centre of Gravity and the Centre of Mass is at the same position.