Romeo takes a uniform 13.7-m ladder and leans it against the smooth (frictionless) wall of the Capulet residence. The ladder's mass is 23.2 kg and the bottom rests on the ground 3.43 m from the wall. When Romeo, whose mass is 70 kg, gets 88.3 percent of the way to the top, the ladder begins to slip. What is the coefficient of static friction between the ground and the ladder?
To determine the coefficient of static friction between the ground and the ladder, we can analyze the forces acting on the ladder.
1. Calculate the weight of the ladder:
The weight (W) of the ladder 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.
In this case, the mass of the ladder is given as 23.2 kg. The acceleration due to gravity is approximately 9.8 m/s². So, the weight of the ladder is W = 23.2 kg * 9.8 m/s².
2. Calculate the normal force exerted on the ladder:
The normal force (N) exerted on the ladder by the ground is equal to the weight of the ladder (since the ladder is in static equilibrium).
So, N = W (weight of the ladder).
3. Calculate the force exerted by Romeo on the ladder:
The force (F) exerted by Romeo on the ladder can be calculated using the formula: F = m * g, where m is the mass of Romeo.
In this case, Romeo's mass is given as 70 kg. So, the force exerted by Romeo on the ladder is F = 70 kg * 9.8 m/s².
4. Calculate the frictional force:
The frictional force (f) acting between the ladder and the ground can be calculated using the formula: f = μ * N, where μ is the coefficient of static friction and N is the normal force.
Since we want to find the coefficient of static friction, we rearrange the formula as: μ = f / N.
In this case, the normal force (N) is equal to the weight of the ladder, which we calculated earlier, and the frictional force (f) is equal to the force exerted by Romeo on the ladder.
5. Calculate the distance to the top of the ladder when it starts to slip:
The distance to the top of the ladder when it starts to slip can be calculated using the formula: d = x * L, where x is the fractional distance to the top where Romeo is, and L is the length of the ladder.
In this case, the length of the ladder is given as 13.7 m, and Romeo gets 88.3% (0.883) of the way to the top. So, the distance to the top of the ladder when it starts to slip is d = 0.883 * 13.7 m.
6. Determine the minimum angle when the ladder starts to slip:
The minimum angle (θ) when the ladder starts to slip can be calculated using the formula: θ = arctan(d / r), where d is the distance to the top of the ladder when it starts to slip and r is the distance from the bottom of the ladder to the wall.
In this case, we have both the values of d and r to calculate the minimum angle.
7. Use the minimum angle to find the coefficient of static friction:
To find the coefficient of static friction (μ), we use the formula: μ = tan(θ). We substitute the value of θ obtained in the previous step to calculate the coefficient of static friction.
Once you have all the necessary values, plug them into the formulas to find the coefficient of static friction between the ground and the ladder.