A particle of weight W is attached by a light inextensible string of length a to a point A on a verticle wall. The particle is supported in equilbrium by a light rigid strut of length b attached to a point B on the wall at a distance a vertically below A. Show that the tension in the string is W and find the thrust in the rod.
To show that the tension in the string is W, we need to analyze the forces acting on the particle in equilibrium.
Let's start by considering the vertical forces. The weight of the particle acts vertically downward, and since the particle is in equilibrium, the tension force in the string must act vertically upward, in order to balance the weight.
Now let's consider the horizontal forces. Since the string is light and inextensible, it can only exert a force along its direction. Therefore, the tension force in the string has no horizontal component, as it cannot pull or push the particle horizontally.
So, we can conclude that the tension in the string is equal to the weight of the particle, W.
Now, let's find the thrust in the rod. To do this, we need to consider the horizontal forces. The only horizontal force acting on the particle is the thrust force exerted by the rod.
We can use trigonometry to determine the horizontal component of the thrust force.
Consider the right-angled triangle formed by the rod, with the hypotenuse being the rod itself, the vertical side is a (distance from A to B), and the horizontal side is b (length of the rod).
Using trigonometric ratios, we have:
cosθ = adjacent/hypotenuse
cosθ = b/a
From this, we can solve for b:
b = a * cosθ
The horizontal component of the thrust force is equal to the horizontal component of the tension force in the string, which is zero. Therefore, the horizontal component of the thrust force is also zero.
So, the thrust in the rod is zero.
In summary:
- The tension in the string is equal to the weight of the particle, W.
- The thrust in the rod is zero.