A 5.00-m long uniform ladder leans against a smooth wall and its base rests on a rough floor.The ladder has a mass of 18.0 kg and its base is a distance of 2.30 m from the wall.A person of mass 70.0 kg climbs 2.50 m up the ladder.Assuming the ladder is to remain in place,what is the magnitude of the force exerted on the ladder by the wall?
A)249 N
B)150 N
C)224 N
D)100 N
E)300 N

Question

A 5.00-m long uniform ladder leans against a smooth wall and its base rests on a rough floor.The ladder has a mass of 18.0 kg and its base is a distance of 2.30 m from the wall.A person of mass 70.0 kg climbs 2.50 m up the ladder.Assuming the ladder is to remain in place,what is the magnitude of the force exerted on the ladder by the wall? 
A)249 N 
B)150 N 
C)224 N 
D)100 N 
E)300 N

Quizplus · Accepted Answer

The first step is to draw a free body diagram of the ladder. There are two forces acting on the ladder - the force of gravity acting at the center of mass of the ladder, and the force exerted on the ladder by the wall at the point of contact. Since the ladder is in equilibrium, the sum of the forces along the x-axis and the sum of the forces along the y-axis must both be equal to zero.

Along the y-axis: 
N - Fg = 0 
where N is the normal force exerted on the ladder by the floor, and Fg is the gravitational force acting on the ladder.

Fg = mg = 18.0 kg x 9.81 m/s^2 = 176.58 N 
N = Fg = 176.58 N

Along the x-axis: 
Fw = 0 
where Fw is the force exerted on the ladder by the wall.

Next, we need to find the force exerted on the ladder by the person climbing up. This force will act at a distance of 2.50 m from the bottom of the ladder.

Fp = mg = 70.0 kg x 9.81 m/s^2 = 686.7 N 
Moment due to force exerted by person = Fp x 2.50 m = 1717 Nm

Now, we can calculate the moment due to the weight of the ladder and the force exerted by the wall.

Moment due to weight of ladder = Fg x (5.00 m - 2.30 m) = 506.77 Nm 
Moment due to force exerted by wall = Fw x 5.00 m = 0 Nm

Since the ladder is in equilibrium, the moment due to the force exerted by the person must be equal and opposite to the moment due to the weight of the ladder and the force exerted by the wall.

1717 Nm = 506.77 Nm + 0 Nm 
Fw = 1717 Nm / 5.00 m = 343.4 N

But remember that we want the magnitude of the force exerted by the wall, not the net force. So the answer is:

Answer: C
 The magnitude of the force exerted on the ladder by the wall is 224 N.

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