A ladder of length 10m and weight 200N leans against a smooth wall such that its at an angle of 60 degrees to the horizontal. A boy of weight 500N stands on d ladder at 2.5m of the way from the lower end of the ladder. Calculate the normal reaction at the wall and the magnitude of the resultant force.

140N

To calculate the normal reaction at the wall and the magnitude of the resultant force, let's break it down step by step.

Step 1: Calculate the gravitational force acting on the ladder:
The weight of the ladder is given as 200N.
The weight of the boy is given as 500N.
The total weight of the ladder and the boy is 200N + 500N = 700N.

Step 2: Resolve the weight of the ladder and the boy into horizontal and vertical components:
The ladder is at an angle of 60 degrees to the horizontal. To find the vertical component (perpendicular to the wall), use the formula: weight × sin(angle).
Vertical component = 700N × sin(60°) = 700N × 0.866 = 606.2N.

To find the horizontal component (parallel to the wall), use the formula: weight × cos(angle).
Horizontal component = 700N × cos(60°) = 700N × 0.5 = 350N.

Step 3: Calculate the normal reaction at the wall:
The normal reaction is the force exerted by the wall on the ladder, perpendicular to the wall.
Since the ladder is in equilibrium, the vertical forces are balanced, and the normal reaction balances the vertical component.
Therefore, the normal reaction is equal to the vertical component: 606.2N.

Step 4: Calculate the magnitude of the resultant force:
The resultant force is the vector sum of the horizontal component (parallel to the wall) and the vertical component (perpendicular to the wall).
To find the magnitude of the resultant force, use the formula: √(horizontal component^2 + vertical component^2).
Magnitude of the resultant force = √(350N^2 + 606.2N^2) = √(122500N^2 + 367944.44N^2) = √490444.44N^2 = 700.85N (approximately).

So, the normal reaction at the wall is 606.2N and the magnitude of the resultant force is approximately 700.85N.

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