A person of mass m2= 85.0 kg is standing on a rung, one third of the way up a ladder of length d= 4.0 m. The mass of the ladder is m1= 15.0 kg, uniformly distributed. The ladder is initially inclined at an angle θ= 40.0∘ with respect to the horizontal. Assume that there is no friction between the ladder and the wall but that there is friction between the base of the ladder and the floor with a coefficient of static friction μs .
(a) Using the equations of static equilibrium for both forces and torque, find expressions for the normal and horizontal components of the contact force between the ladder and the floor, and the normal force between the ladder and the wall. Consider carefully which point to use for computing the torques. Determine the magnitude of the frictional force (in N) between the base of the ladder and the floor below.
f_s=418.5
(b) Find the magnitude for the minimum coefficient of friction between the ladder and the floor so that the person and ladder does not slip.
mu_s= 0.426
(c) Find the magnitude Cladder,ground (in N) of the contact force that the floor exerts on the ladder. Remember, the contact force is the vector sum of the normal force and friction.
UNANSWERED
Find the direction of the contact force that the floor exerts on the ladder. i.e. determine the angle α (in radians) that the contact force makes with the horizontal to indicate the direction.
UNANSWERED
how did you get part a and b???
@jessica
b) Find Mu_s first
Mu_s= (m_p/3 + m_l/2) *cotan (theta)/(m_p+m_l)
a) Force = Mu_s*g(m_p+m_l)
http://web.mit.edu/8.01t/www/materials/InClass/IC_Sol_W09D3-2.pdf
To find the magnitude of the contact force that the floor exerts on the ladder (Cladder,ground), we need to consider the vertical and horizontal components of the contact force.
First, let's consider the vertical component. Since the ladder is in equilibrium, the sum of the vertical forces must be zero. Thus, the normal force N between the ladder and the floor cancels out the weight of both the ladder and the person standing on it.
The weight of the ladder is given by w1 = m1 * g, where m1 is the mass of the ladder and g is the acceleration due to gravity. Similarly, the weight of the person is w2 = m2 * g. Since the ladder is uniformly distributed, its weight acts at its center of mass, which is at a distance of d/2 = 4.0 m / 2 = 2.0 m from the base.
The vertical component of the contact force is then given by N = w1 + w2, since these weights act downwards.
Next, let's consider the horizontal component. Since the ladder is in equilibrium, the sum of the horizontal forces must be zero. This means that the frictional force between the base of the ladder and the floor cancels out the horizontal component of the contact force.
Using θ = 40.0°, we can determine the length of the ladder along the incline as d1 = d * cos(θ). The horizontal component of the contact force is then given by Fh = μs * N, where μs is the coefficient of static friction.
Now, to find the magnitude of the frictional force (in N) between the base of the ladder and the floor, we can use the equation:
Frictional force = μs * N = μs * (w1 + w2)
Plugging in the given values:
μs = 0.426
m1 = 15.0 kg
m2 = 85.0 kg
d = 4.0 m
θ = 40.0°
μs * (w1 + w2) = 0.426 * ((15.0 kg * g) + (85.0 kg * g))
Now, you can calculate the frictional force.
(a) The magnitude of the frictional force between the base of the ladder and the floor is given as f_s = 418.5 N.
Moving on to the remaining unanswered questions:
(b) To find the minimum coefficient of friction (μs) between the ladder and the floor so that the ladder and person do not slip, we can use the equation:
μs(min) = tan(θ)
Substituting the given value of θ = 40.0°, you can calculate the minimum coefficient of friction.
(c) To find the magnitude of the contact force (Cladder,ground) that the floor exerts on the ladder, we can use the equation:
Cladder,ground = N + f_s
(d) To find the direction of the contact force, we need to determine the angle α that the contact force makes with the horizontal. This can be calculated as:
α = tan^(-1)(Fh/N)
Substituting the calculated values of Fh and N, you can find the angle α in radians.