A 150-kg ladder leans against a smooth wall, making an angle of 30 degrees with the floor. The centre of gravity of the ladder is one-third the way up from the bottom. How large a horizontal force must the floor provide if the ladder is not to slip?
To determine the horizontal force required for the ladder not to slip, we need to analyze the forces acting on the ladder and the conditions for equilibrium.
1. Begin by drawing a diagram of the ladder leaning against the wall. Label the important angles and lengths.
|
|
|\
| \
| \ height = h
| \
| \
_______________|_____\_________________
| |
| |
| |
ground wall
2. Identify the forces acting on the ladder.
a. Weight of the ladder: This force acts downwards, vertically. Its magnitude is given as the product of the mass of the ladder (150 kg) and the acceleration due to gravity (9.8 m/s^2). So, the weight of the ladder is W = 150 kg * 9.8 m/s^2.
b. Normal force: This force acts perpendicular to the wall and supports the ladder's weight. It is equal in magnitude but opposite in direction to the vertical component of the weight.
c. Frictional force: This force acts horizontally, preventing the ladder from slipping. Its magnitude is unknown and needs to be determined.
d. Horizontal force: This force is exerted by the floor and opposes the horizontal component of the weight and frictional force.
3. Analyze the conditions for equilibrium.
a. Since the ladder is not slipping, the net force acting in the horizontal direction must be zero. This means the horizontal component of the weight (W_x) should be balanced by the horizontal frictional force (f).
b. The net force acting in the vertical direction must also be zero for equilibrium. Since the ladder is smooth and there is no vertical force applied, the vertical component of the weight (W_y) is balanced by the normal force.
4. Calculate the magnitudes of the horizontal and vertical components of the weight.
a. The angle between the ladder and the floor is given as 30 degrees. We can use this angle to calculate the horizontal component of the weight, W_x, using trigonometric functions.
W_x = W * cos(30)
b. The vertical component of the weight, W_y, is calculated using the angle between the ladder and the floor as:
W_y = W * sin(30)
5. Determine the magnitude of the normal force.
Since the ladder is in equilibrium, the vertical component of the weight, W_y, is balanced by the normal force. Therefore, the normal force is equal to W_y.
6. Find the frictional force.
Since the ladder is not slipping, the frictional force is given by the product of the normal force and the coefficient of static friction (μ), which is unknown.
f = μ * N
7. Set up an equation for equilibrium in the horizontal direction.
The horizontal components of the weight (W_x) and the frictional force (f) should balance each other to prevent slipping.
W_x = f
8. Substitute the known values into the equation to solve for the unknown frictional force (f).
W_x = f
W * cos(30) = μ * N
9. Substitute the vertical component of the weight, W_y, for the normal force (N).
W * cos(30) = μ * W_y
10. Substitute the known value for the vertical component of the weight.
W * cos(30) = μ * W * sin(30)
11. Solve for the unknown coefficient of static friction (μ).
μ = cos(30) / sin(30)
12. Now that we have the coefficient of static friction (μ), we can find the frictional force (f) by multiplying it by the normal force.
f = μ * N
13. Finally, to calculate the required horizontal force exerted by the floor, add the frictional force (f) to the horizontal component of the weight (W_x).
Required horizontal force = f + W_x