An 8.5 m extension ladder of mass 24 kg is propped up against a wall, touching at a point 8.0 m above the level ground. A man of mass 75 kg climbs 2.0 m up the ladder to repair a window. The ladder rests against a frictionless wall, but the ground has friction. Determine the magnitude and direction of all forces on the ladder.
Normal force exerted by the ground.
(Answer: 970N)
Frictional force exerted by ground.
(Answer: 104N)
Force exerted by wall.
(Answer: 104N)
Force exerted by man.
(Answer: 735N)
Force exerted by weight of ladder.
(Answer: 235N)
Answers are provided above. Please show all work on how to get the answers.
To determine the magnitude and direction of all forces on the ladder, we need to analyze the forces acting on the ladder. Here's a step-by-step breakdown of the problem:
1. Determine the weight of the ladder:
The weight of an object is given by the formula W = m * g, where m is the mass of the object and g is the acceleration due to gravity (9.8 m/s²).
In this case, the mass of the ladder is 24 kg, so the weight of the ladder is W_ladder = 24 kg * 9.8 m/s² = 235.2 N.
Therefore, the force exerted by the weight of the ladder is 235.2 N, acting vertically downward.
2. Determine the weight of the man:
Similar to the ladder, the weight of the man can be calculated using the same formula W = m * g.
The mass of the man is 75 kg, so the weight of the man is W_man = 75 kg * 9.8 m/s² = 735 N.
Therefore, the force exerted by the weight of the man is 735 N, acting vertically downward.
3. Determine the normal force exerted by the ground:
The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, it will counteract the weight of the ladder and the weight of the man.
The normal force acts perpendicular to the surface, which in this case is vertically upward.
Since the ladder is at rest, the net vertical force must be zero.
Therefore, the normal force exerted by the ground is equal to the sum of the weight of the ladder (235.2 N) and the weight of the man (735 N),
Normal force = 235.2 N + 735 N = 970.2 N, which can be rounded to 970 N.
4. Determine the frictional force exerted by the ground:
Since the ladder is propped against the wall, it is in equilibrium, meaning the sum of all horizontal forces acting on it must be zero.
The only horizontal force acting on the ladder is the frictional force exerted by the ground.
The force of friction can be calculated using the formula f_friction = μ * f_normal, where μ is the coefficient of friction.
Since the ground has friction, we need to know the coefficient of friction in order to calculate the frictional force.
Without the coefficient of friction value, it is not possible to determine the exact magnitude of the frictional force.
5. Determine the force exerted by the wall:
In this case, the ladder rests against a frictionless wall, which means there is no frictional force acting between the ladder and the wall.
Therefore, the force exerted by the wall is zero.
To summarize:
- The normal force exerted by the ground is 970 N, acting vertically upward.
- The frictional force exerted by the ground cannot be determined without the coefficient of friction value.
- The force exerted by the wall is zero.
- The force exerted by the man is 735 N, acting vertically downward.
- The force exerted by the weight of the ladder is 235.2 N, acting vertically downward.
a) 24kg * 9.8m/s^2 + 75kg * 9.8m/s^2 = 970N
b) A = sqrt[(8.5m)^2 - (8m)^2] = 2.87m
CC = 2m
CC / AC = 2m/B = 8m / 2.87m
AC= 2m * 2.87 / 8m
AC = 0.7175m
BA = 8.5m / 2m = 4.25m
AC / BA = AO / AD
AC / 4.25m = 2.87m / 8.5m
AC2 = 2.87m * 4.25m / 8.5m
AC2 = 1.435m
F2AC1 + F1AC2 - Ff * 8m = 0
Ff = 75kg * 9.8m/s^2 * 2.5m + 24kg * 9.8m/s^2 * 0.41m / 8m
Ff = (735N * 0.7175m + 235.2N * 1.435m) / 8m
Ff = 104N
c) Force exerted by wall = Frictional force exerted by ground.
Force by wall = 104NN
d) Force of man = 75kg * 9.8m/s^2 = 735N
e) Force of ladder = 24kg * 9.8m/s^2 = 235N