A ladder AB of mass 20 kg rest on smooth horizontal ground and leans against a sooth vertical wall. The inclination of the ladder to the horizontal is 60°. The ladder is kept in position by a horizontal force P applied to the bottom of the ladder.
Find the value of P and the reactions at the wall and the ground.

Question

A ladder AB of mass 20 kg rest on smooth horizontal ground and leans against a sooth vertical wall. The inclination of the ladder to the horizontal is 60°. The ladder is kept in position by a horizontal force P applied to the bottom of the ladder.
Find the value of P and the reactions at the wall and the ground.

Accepted Answer

∻Solution∻ ‖ a ⋮ Identify the forces acting on the ladder: The weight of the ladder ($W = mg = 20 \, ext{kg} imes 9.81 \, ext{m/s}^2 = 196.2 \, ext{N}$) acts downward at the center of the ladder. The horizontal force $P$ acts at the bottom of the ladder, and the normal force $N$ acts upward at the ground. The wall exerts a reaction force $R$ horizontally at the top of the ladder. ‖ ‖ b ⋮ Apply the equilibrium conditions: For the ladder to be in static equilibrium, the sum of the forces in both the horizontal and vertical directions must be zero: $$\sum F_x = 0$$ and $$\sum F_y = 0$$. Thus, we have: $$R - P = 0$$ (1) and $$N - W = 0$$ (2). From (2), we find that $$N = W = 196.2 \, ext{N}$$. ‖ ‖ c ⋮ Calculate the torques about the base of the ladder: The torque due to the weight of the ladder is given by $$ au_W = W \cdot d_W$$, where $d_W$ is the horizontal distance from the base to the center of the ladder. The length of the ladder can be calculated using trigonometry: $$L = \frac{h}{\sin(60^\circ)}$$, where $h$ is the vertical height. The horizontal distance to the center is $$d_W = \frac{L}{2} \cdot \cos(60^\circ)$$. The torque due to the reaction force at the wall is $$ au_R = R \cdot L \cdot \sin(60^\circ)$$. Setting the torques equal gives us: $$W \cdot d_W = R \cdot L \cdot \sin(60^\circ)$$. ‖ ‖ d ⋮ Substitute the known values into the torque equation and solve for $P$: After calculating $L$ and $d_W$, we can find $R$ using the equilibrium condition (1). Finally, we can express $P$ in terms of $R$: $$P = R$$. ‖ ∻Answer∻ ⚹ P = R, N = 196.2 N, R = 98.1 N ⚹ ∻Key Concept∻ ⚹ Static Equilibrium: For an object to be in static equilibrium, the net force and net torque acting on it must be zero. This involves balancing forces and calculating torques about a pivot point. Equations: $$\sum F = 0$$ and $$\sum au = 0$$. ⚹ ∻Explanation∻ ⚹ By analyzing the forces and torques acting on the ladder, we can determine the necessary horizontal force $P$ and the reactions at the wall and ground to maintain equilibrium. ⚹

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