A uniform ladder of length 20.0m and weight 750 N is propped up against a smooth vertical wall with its lower end on a rough horizontal surface. The coefficient of friction between the ladder and this horizontal surface is 0.40.
(a) Draw a diagram for the ladder showing all the forces acting on the ladder.
(b) Work out and add the numerical values of each force clearly showing your justification in each case.
(c) Hence, calculate a value for the angle between the ladder and the wall if the ladder just remains in stable equilibrium.
(a) Diagram:
```
/
/
/|
/ |
/ |
/ | Ladder (20.0m, 750 N)
/ |
/ |
/theta |
|
|________________
Horizontal surface
```
Forces acting on the ladder:
1. Weight of the ladder (W) = 750 N (downward)
2. Normal force from the horizontal surface (N) - perpendicular to the surface
3. Frictional force from the horizontal surface (f) - parallel to the surface
4. Normal force from the wall (Fn) - perpendicular to the wall
5. Force of tension (T) - along the ladder, pushing it against the wall
(b) Numerical values of each force:
1. Weight of the ladder (W) = 750 N (downward)
2. Normal force from the horizontal surface (N) - Since the ladder is in equilibrium, the normal force is equal in magnitude and opposite in direction to the weight. Therefore, N = 750 N (upward)
3. Frictional force from the horizontal surface (f) - The frictional force can be calculated using the formula: f = coefficient of friction * normal force. In this case, the coefficient of friction is 0.40, and the normal force is 750 N. Therefore, f = 0.40 * 750 N = 300 N (opposite to the direction of T)
4. Normal force from the wall (Fn) - Since the ladder is in equilibrium, the normal force is perpendicular to the wall. Therefore, Fn = 750 N (perpendicular to the wall)
5. Force of tension (T) - This is the force that keeps the ladder in stable equilibrium. To calculate T, we need to consider the forces acting along the ladder. The vertical component of T counters the weight of the ladder, while the horizontal component of T counters the frictional force. Using trigonometry, we can relate T to the angle between the ladder and the wall (theta). Therefore, T = Weight of the ladder / cos(theta) = 750 N / cos(theta)
(c) Calculation for the angle theta:
To find the angle theta, we need to set up an equation and solve for it.
T = Weight of the ladder / cos(theta)
T = 750 N / cos(theta)
Since the ladder is in stable equilibrium, the vertical component of T equals the weight of the ladder (W):
T * sin(theta) = W
T * sin(theta) = 750 N
Using the above two equations, we can find the angle theta:
750 N / cos(theta) * sin(theta) = 750 N
Dividing both sides of the equation by 750 N:
1 / cos(theta) * sin(theta) = 1
Multiplying both sides of the equation by cos(theta):
sin(theta) = cos(theta)
Taking the inverse sine of both sides of the equation:
theta = arcsin(cos(theta))
Using trigonometric calculations, we find that theta is approximately 60 degrees. Therefore, the angle between the ladder and the wall for stable equilibrium is approximately 60 degrees.
(a) Diagram:
```
|\
| \
F | \ Fg
| \
| \
| \
----------------|------\---------
| | /
| | /
| - | - /
| | Fr
| |
|<---- Ff ---->|
```
Explanation:
- Fg is the weight of the ladder (750 N) acting vertically downwards at its center of gravity.
- Fr is the normal reaction force exerted by the rough horizontal surface on the ladder. It acts perpendicular to the surface.
- Ff is the frictional force exerted by the horizontal surface on the ladder. It acts parallel to the surface and opposes the motion.
- F is the force applied by the wall on the ladder. It acts perpendicular to the wall.
(b) Numerical Values of Forces:
- Fg = 750 N (given in the problem)
- Fr = Fg = 750 N (since the ladder is in equilibrium vertically)
- Ff = μ * Fr = 0.40 * 750 N = 300 N (using the given coefficient of friction)
- F = sqrt(Ff^2 + Fr^2) = sqrt(300^2 + 750^2) ≈ 806.23 N (using Pythagorean theorem)
Explanation:
- The normal reaction force (Fr) is equal in magnitude and opposite in direction to the weight of the ladder (Fg) since the ladder is in equilibrium vertically.
- The frictional force (Ff) is calculated by multiplying the coefficient of friction (μ) with the normal reaction force (Fr), as given in the problem.
- The force (F) applied by the wall on the ladder is calculated using the Pythagorean theorem because it acts at an angle with the vertical direction, opposing the combination of frictional force (Ff) and the vertical force (Fr).
(c) Calculation of Angle:
- cosθ = Ff / F = 300 N / 806.23 N ≈ 0.372
- θ = arccos(0.372) ≈ 68.71 degrees
Hence, the value of the angle between the ladder and the wall for the ladder to remain in stable equilibrium is approximately 68.71 degrees.