a uniform ladder 16 m long weighing 80 N leans against a smooth vertical wall, making 30 degree with the wall. Find the thrust of the wall, and the force of the point on the ground.
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To find the thrust of the wall and the force of the point on the ground, we can use the principles of equilibrium. Since the ladder is in equilibrium, the sum of the forces acting in both the vertical and horizontal directions must be zero.
Let's start by resolving the forces acting on the ladder.
1. Resolving forces in the vertical direction:
The weight of the ladder acts vertically downward and can be split into two components:
- The component parallel to the wall, which provides the thrust (T) of the wall.
- The component perpendicular to the wall, which gets balanced by the normal reaction force (N) at the point of contact with the ground.
The weight of the ladder (W) can be calculated using the formula W = m * g, where g is the acceleration due to gravity and m is the mass of the ladder. Given that the weight of the ladder is 80 N, we can assume the mass of the ladder to be 80 N / 9.8 m/s^2 ≈ 8.16 kg.
The component of the weight parallel to the wall is W * cosθ, where θ is the angle the ladder makes with the wall. Therefore, the thrust of the wall (T) is given by T = W * cosθ.
The component of the weight perpendicular to the wall is W * sinθ. Therefore, the normal reaction force (N) is given by N = W * sinθ.
2. Resolving forces in the horizontal direction:
Since the ladder is smooth, there is no friction acting horizontally. Hence, the only horizontal force is the force of the ground (F) at the point of contact.
Now we can substitute the values into the equations:
T = W * cosθ
= (8.16 kg * 9.8 m/s^2) * cos30°
≈ 79.97 N ≈ 80 N (approx.)
N = W * sinθ
= (8.16 kg * 9.8 m/s^2) * sin30°
≈ 41.96 N ≈ 42 N (approx.)
Therefore, the thrust of the wall is approximately 80 N, and the force at the point of contact on the ground is approximately 42 N.