A 16.7 kg person climbs up a uniform ladder with negligible mass. The upper end of the ladder rests on a frictionless wall. The bottom of the ladder rests on a floor with a rough surface where the coefficient of static friction is 0.2. The angle between the horizontal and the ladder is 0. The person wants to climb up the ladder a distance of 1.2 m along the ladder from the ladder's foot. What is the minimum angle min (between the horizontal and the ladder) so that the person can reach a distance of 1.2 m without having the ladder slip? The acceleration of gravity is 9.8 m/s Answer in units of °.

Question

A 16.7 kg person climbs up a uniform ladder with negligible mass. The upper end of the ladder rests on a frictionless wall. The bottom of the ladder rests on a floor with a rough surface where the coefficient of static friction is 0.2. The angle between the horizontal and the ladder is 0. The person wants to climb up the ladder a distance of 1.2 m along the ladder from the ladder's foot. What is the minimum angle min (between the horizontal and the ladder) so that the person can reach a distance of 1.2 m without having the ladder slip? The acceleration of gravity is 9.8 m/s Answer in units of °.

Answer 1

To prevent the ladder from slipping, the horizontal component of the force exerted by the person (F_h) should be equal to or greater than the force of static friction (f_r) between the ladder and the floor.

To find the horizontal component of the force exerted by the person, we can use the equation F_h = F * cosθ, where F is the total force exerted by the person and θ is the angle between the force and the ladder.

First, we need to find the total force exerted by the person by using the equation F = m * g, where m is the mass of the person and g is the acceleration due to gravity.

F = 16.7 kg * 9.8 m/s^2
F ≈ 163.66 N

Next, we can substitute the value of F into the equation F_h = F * cosθ to get an equation for F_h in terms of θ.

F_h = 163.66 N * cosθ

Now, we can set up an expression for the force of static friction (f_r) using the equation f_r = μ * f_n, where μ is the coefficient of static friction and f_n is the normal force acting on the ladder.

f_n = m * g
f_n = 16.7 kg * 9.8 m/s^2
f_n ≈ 163.66 N

f_r = 0.2 * 163.66 N
f_r ≈ 32.73 N

Since the force of static friction (f_r) must be less than or equal to the horizontal component of the force exerted by the person (F_h), we can set up the following inequality.

F_h ≥ f_r
163.66 N * cosθ ≥ 32.73 N

Now, we can solve this inequality for θ.

cosθ ≥ 32.73 N / 163.66 N
cosθ ≥ 0.2

To find the minimum angle (min) between the horizontal and the ladder, we need to find the inverse cosine of 0.2.

min = cos^(-1)(0.2)
min ≈ 78.46°

Therefore, the minimum angle min (between the horizontal and the ladder) is approximately 78.46° so that the person can reach a distance of 1.2 m without having the ladder slip.