A 6m long uniform ladder which has a mass of 25kg leans with its upper end against a smoorth vertical wall and its lower end on rough horizontal ground. The base of the ladder is 3m from the wall and a man of mass 70kg is two thirds of the way up the ladder.
calculate frictional force between the ladder and the ground
determine the magnitude and direction of the resultant force acting on the base of the ladder
use this sample problem. Go through it first.
http://www.jiskha.com/display.cgi?id=1273284765
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F=M
70*9.8=686
25*9.8=245
686+245=921
f=931
still don't know about the magnitude
magnitude: you have friction force and weight on the ladder base. They are at 90 degrees. Add them
so its 931^2-29.4^2
= sqrt so its 930.5 is the magnitude?
i don't think either is right
To calculate the frictional force between the ladder and the ground, we need to first consider the forces acting on the ladder.
1. Weight of the ladder: The weight of the ladder can be calculated by multiplying the mass (25kg) with the acceleration due to gravity (9.8 m/s^2).
Weight of the ladder = 25 kg * 9.8 m/s^2 = 245 N (downwards)
2. Weight of the man: The weight of the man can be calculated in the same way, by multiplying his mass (70kg) with the acceleration due to gravity.
Weight of the man = 70 kg * 9.8 m/s^2 = 686 N (downwards)
3. Normal force: The normal force exerted by the ground on the ladder counters the weight of the ladder and the man. Since the ladder is in equilibrium, the normal force is equal in magnitude but opposite in direction to the sum of the weights.
Normal force = 245 N + 686 N = 931 N (upwards)
4. Frictional force: The frictional force between the ladder and the ground opposes the tendency of the ladder to slip. It acts parallel to the surface of contact between the ladder and the ground. The maximum frictional force can be calculated using the equation:
Frictional force = coefficient of friction * normal force
The coefficient of friction depends on the nature of the contact surface between the ladder and the ground. Since the surface is described as rough, we can assume a higher coefficient of friction, such as 0.6 (representing a high frictional force).
Frictional force = 0.6 * 931 N = 558.6 N (opposes the motion of the ladder)
To determine the magnitude and direction of the resultant force acting on the base of the ladder, we need to consider the horizontal and vertical components of all the forces acting on it.
1. Horizontal forces:
- Frictional force acts in the opposite direction to the ladder's tendency to slip.
- There is also a horizontal component of the weight of the man, which can be calculated as:
Horizontal component of man's weight = (70 kg * 9.8 m/s^2) * (2/3) * cos(theta), where theta is the angle of the ladder from the vertical.
Using trigonometry, we can determine that theta = arccos(4/5) (approximately 36.87 degrees).
Horizontal component of man's weight = (70 kg * 9.8 m/s^2) * (2/3) * cos(36.87 degrees) = 343.2 N (opposite to the frictional force)
The horizontal forces acting on the base of the ladder are:
Resultant horizontal force = - Frictional force + Horizontal component of man's weight
= - 558.6 N + (-343.2 N)
= -901.8 N (negative sign indicates opposite direction)
2. Vertical forces:
The vertical forces acting on the base of the ladder are:
Resultant vertical force = Weight of the ladder + Weight of the man - Normal force
= 245 N + 686 N - 931 N
= 0 N (since the ladder is in equilibrium vertically)
Therefore, the magnitude and direction of the resultant force acting on the base of the ladder are:
Magnitude = sqrt((Resultant horizontal force)^2 + (Resultant vertical force)^2)
= sqrt((-901.8 N)^2 + (0 N)^2)
= 901.8 N
Direction: Since the resultant vertical force is zero, the resultant force acts horizontally. The negative sign indicates that the resultant force is acting in the opposite direction to the frictional force, i.e., towards the wall against which the ladder is leaning.