Romeo takes a uniform 11 m ladder and leans it against the smooth wall of the Capulet residence. The ladder's mass is 18.0 kg, and the bottom rests on the ground 2.8 m from the wall. When Romeo, whose mass is 79 kg, gets 90% of the way to the top, the ladder begins to slip. What is the coefficient of static friction between the ground and the ladder?
physics
To find the coefficient of static friction between the ground and the ladder, we need to analyze the forces acting on the ladder and determine the conditions for it to start slipping.
Let's consider the forces acting on the ladder:
1. Weight of the ladder (W_ladder): The weight acts vertically downward and can be calculated using the formula W = m * g, where m is the mass of the ladder and g is the acceleration due to gravity.
W_ladder = 18.0 kg * 9.8 m/s^2 = 176.4 N
2. Weight of Romeo (W_Romeo): Since Romeo is on the ladder, his weight also acts vertically downward. We can calculate it using the same formula: W = m * g, where m is Romeo's mass.
W_Romeo = 79 kg * 9.8 m/s^2 = 774.2 N
3. Normal force (N): The normal force acts perpendicular to the inclined ladder and opposes the weight. Since the ladder is resting against the wall, there is no vertical acceleration, and the net force in the vertical direction is zero. Therefore,
N = W_ladder + W_Romeo = 176.4 N + 774.2 N = 950.6 N
4. Frictional force (f): The frictional force acts parallel to the ground and opposes the sliding or slipping motion of the ladder. Just before slipping, the frictional force reaches its maximum value, which is given by the equation f = μ * N, where μ is the coefficient of static friction.
Since we want to find the coefficient of static friction, we rearrange the equation to solve for μ:
μ = f / N
Now, we need to find the maximum frictional force. At the point of slipping, Romeo is 90% of the way up the ladder. So, the distance from the center of mass of Romeo to the bottom end of the ladder is:
d = 0.9 * 11 m = 9.9 m
To calculate the torque around the bottom end of the ladder, we can use the relation:
torque = f * d = I * α
where torque is the product of the friction and the lever arm distance d, I is the moment of inertia, and α is the angular acceleration. Since the ladder is treated as a uniform rod, we have:
I = (1/3) * m * L^2
where m is the mass of the ladder and L is the length of the ladder.
I = (1/3) * 18.0 kg * (11 m)^2 = 726 kg * m^2
Since the ladder is just at the point of slipping, the static frictional force provides sufficient torque to keep the ladder in equilibrium. Thus, the torque equation becomes:
f * d = I * α
We know that α can be calculated using the equation:
α = a / R
where a is the angular acceleration and R is the distance from the center of mass to the bottom end of the ladder:
R = L / 2 = 11 m / 2 = 5.5 m
Now, we can substitute the values and solve for f:
f * 9.9 m = 726 kg * m^2 * (a / 5.5 m)
Simplifying the equation gives us:
f = (726 kg * m^2 * a) / (5.5 m * 9.9 m)
Next, we need to find the angular acceleration (a). For an object rolling without slipping, the angular acceleration is related to the linear acceleration (gained by Romeo) by the equation:
a = α * R
Substituting this expression for α in the equation for f gives:
f = (726 kg * m^2 * a) / (5.5 m * 9.9 m) = (726 kg * m^2 * a) / (5.445 m^2)
Since Romeo is 90% of the way to the top, and the total height of the ladder is 11 m, Romeo has climbed a distance:
h = 0.9 * 11 m = 9.9 m
Therefore, the linear acceleration a can be determined using the kinematic equation:
h = (1/2) * a * t^2
where t is the time it takes for Romeo to reach that height. We can solve this equation for a:
a = (2h) / t^2 = (2 * 9.9 m) / t^2
Now, we need to find the time t. To climb 90% of the way, Romeo covers a distance of 0.9 * 11 m. Using the equation of motion:
s = ut + (1/2) * a * t^2
where u is the initial speed (0 since Romeo starts from rest), and s is the distance covered, we can solve for t:
0.9 * 11 m = 0 * t + (1/2) * a * t^2
0.9 * 11 m = (1/2) * a * t^2
Simplifying the equation gives:
t^2 = (0.9 * 11 m * 2) / a
Now we can substitute the value of t^2 in the equation for a:
a = (2 * 9.9 m) / [(0.9 * 11 m / a) * 2] = (2 * 9.9 m * a) / (0.9 * 22 m)
The a term cancels out, leaving:
a = (2 * 9.9 m) / (0.9 * 22 m) = 1.1818 m/s^2
Substituting this value of acceleration into the equation for f:
f = (726 kg * m^2 * 1.1818 m/s^2) / (5.445 m^2)
= 158.8 N
Finally, we can find the coefficient of static friction by dividing f by N:
μ = f / N = 158.8 N / 950.6 N = 0.1669
Therefore, the coefficient of static friction between the ground and the ladder is approximately 0.1669.