A person with mass 70 kg is climbing a ladder (length 6m, mass 20 kg) that leans against a smooth wall (no friction). A frictional force fs between the ladder and the floor keeps it from slipping. The angle between the ladder and the wall is 30 degrees.
What is the magnitude of fs so that the person can reach the top fo the ladder?
The vertical force on the ladder at the bottom is (70 + 20)g = 90 g = 882 N.
There is a clockwise torque about the wall support point due to this force, and it equals 882N*6 sin30 = 2646 N*m
There is also a counterclockwise torque about the same point, equal to
fs*6sin30 + 20g*6 cos30 + 70g*x cos30
where x is the distance of the man from the top of the ladder.
The largest value of fs needed to obtain equilibrium is required when x is least, e.g., when x = 0 and the man is at the top of the ladder. At that time,
2646 N = 3 fs + 1018N
Solve for fs
To find the magnitude of the frictional force (fs) required for the person to reach the top of the ladder, we need to consider the forces acting on the ladder.
1. The weight of the person (mg = mass × gravitational acceleration):
- Weight of the person = 70 kg × 9.8 m/s^2 = 686 N (downwards)
2. The weight of the ladder:
- Weight of the ladder = 20 kg × 9.8 m/s^2 = 196 N (downwards)
3. The normal force acting on the ladder (N):
- Normal force = weight of the ladder + weight of the person = 196 N + 686 N = 882 N (upwards)
4. The force of friction (fs) opposing the sliding motion:
- The maximum force of static friction (fs) between two surfaces is given by the equation: fs = μ * N
where μ is the coefficient of static friction, and N is the normal force.
Since the ladder is in equilibrium (no vertical acceleration), the sum of the vertical forces must be zero:
Sum of vertical forces = weight of the person + weight of the ladder + fs - N = 0
This can be rearranged to solve for fs:
fs = N - (weight of the person + weight of the ladder)
Now, let's calculate the value of fs:
fs = 882 N - (686 N + 196 N)
fs = 882 N - 882 N
fs = 0 N
Therefore, the magnitude of the frictional force (fs) required for the person to reach the top of the ladder is zero. No frictional force is needed in this scenario.