AB ia a uniform rod of length 120 cm and weight 18kg.wt it rotates about a hinge at its end A. its end B is pulled by a light string whose other end is tied to point C over A. the rod is in equilibrium when it inclines to the horizontal at 30degree if the reaction of the hinge is horizontal, find the magnitude of the tension in the string and the reaction of the hinge
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To find the magnitude of the tension in the string and the reaction of the hinge, we need to analyze the forces acting on the rod when it is in equilibrium.
Let's denote the magnitude of the tension in the string as T and the reaction of the hinge as R. We can break down the forces acting on the rod as follows:
1. Weight of the rod (W): The weight of the rod acts vertically downwards from its center of mass. The weight can be calculated using the formula W = m * g, where m is the mass of the rod and g is the acceleration due to gravity. In this case, the weight of the rod is given as 18 kg.
2. Tension in the string (T): The tension in the string acts horizontally towards point B and is responsible for keeping the rod in equilibrium. This tension force can be resolved into its vertical and horizontal components.
3. Reaction of the hinge (R): The reaction of the hinge acts vertically upwards to balance the weight of the rod and keep it in equilibrium. Since the reaction of the hinge is horizontal, it does not contribute to the vertical equilibrium of the rod.
When the rod is inclined at 30 degrees to the horizontal, the vertical component of the tension in the string must balance the weight of the rod, while the horizontal component of the tension balances the horizontal reaction of the hinge.
Now, let's calculate the forces involved:
Vertical equilibrium: The vertical component of the tension in the string (T * sinθ) must balance the weight of the rod (W). Here, θ is the angle of inclination, which is 30 degrees.
T * sinθ = W
Substituting the values, we have:
T * sin(30 degrees) = 18 kg * 9.8 m/s^2
T * 0.5 = 176.4 N
T = 176.4 N / 0.5 = 352.8 N
So, the magnitude of the tension in the string is 352.8 N.
Horizontal equilibrium: The horizontal component of the tension in the string (T * cosθ) must balance the horizontal reaction of the hinge (R).
T * cosθ = R
Since the reaction of the hinge is horizontal, it does not depend on the weight or the angle of inclination. Thus, T * cos(30 degrees) = R.
Substituting the values, we have:
T * 0.866 = R
T = 352.8 N
R = 352.8 N * 0.866 = 305.6 N
So, the magnitude of the reaction of the hinge is 305.6 N.
Therefore, the magnitude of the tension in the string is 352.8 N and the magnitude of the reaction of the hinge is 305.6 N.