A ladder, 5.0 m long, leans against a frictionless wall at a
point 4.0 m above the ground. A painter is climbing up the
ladder. The mass of the ladder is 12.0 kg and the mass of the
painter is 60.0 kg.
The ladder begins to slip at its base when the painter is 70 %
of the way up the length of the ladder. What is the coefficient
of static friction between the ladder and the floor?
(The base of the ladder is 3 meters from the wall)
please help asap
To find the coefficient of static friction between the ladder and the floor, we need to analyze the forces acting on the ladder.
We can start by drawing a free-body diagram of the ladder:
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5.0 m |||
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Let's consider the forces acting on the ladder.
1. Weight: The ladder has a weight due to its mass. The weight acts vertically downward, through the center of mass of the ladder and painter. We can calculate the total weight using the sum of individual weights.
Weight of ladder = mass of ladder * acceleration due to gravity (g)
= 12.0 kg * 9.8 m/s²
= 117.6 N
Weight of painter = mass of painter * g
= 60.0 kg * 9.8 m/s²
= 588.0 N
Total weight = Weight of ladder + Weight of painter
2. Normal Force: The wall provides a normal force perpendicular to the wall. As the ladder is leaning against the wall, the normal force acts in the opposite direction of the weight. The normal force cancels out the vertical component of the weight.
3. Frictional Force: The frictional force acts horizontally opposite to the direction of any impending or actual relative motion between the ladder and the floor. In this case, the ladder is just on the verge of slipping, so the frictional force is at its maximum value and equal to the product of the normal force and the coefficient of static friction (μ).
Now, we need to find the normal force. Since the ladder is in equilibrium (not accelerating in the vertical direction), the sum of vertical forces must be zero.
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| Normal Force || | |
| (opposite to weight) || | |
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5.0 m |||
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The normal force can be found using the following equation:
Normal force + Weight of ladder + Weight of painter = 0
Since the normal force is opposite to the weight, we have:
Normal force = -(Weight of ladder + Weight of painter)
Substituting the values:
Normal force = -(117.6 N + 588.0 N)
= -705.6 N (Note: The negative sign indicates the opposite direction to the weight)
Now, we can calculate the frictional force. The maximum static frictional force can be given as:
Frictional force (max) = Normal force * coefficient of static friction
We need to find the maximum static frictional force when the ladder is on the verge of slipping. In this case, the force acting downward is 70% of the total weight (since the painter is 70% of the way up), and the remaining part (30%) is acting upward.
Frictional force (max) = 70% * (Weight of ladder + Weight of painter)
Substituting the values:
Frictional force (max) = 0.7 * (117.6 N + 588.0 N)
= 0.7 * 705.6 N
= 493.92 N
Now, we can find the coefficient of static friction (μ) by dividing the frictional force by the normal force:
Coefficient of static friction (μ) = Frictional force (max) / Normal force
Substituting the values:
Coefficient of static friction (μ) = 493.92 N / (-705.6 N)
= -0.7
Since the coefficient of friction cannot be negative, the negative sign indicates an opposite direction to the assumed direction. Therefore, we take the absolute value:
Coefficient of static friction (μ) = |-0.7|
= 0.7
So, the coefficient of static friction between the ladder and the floor is 0.7.