An arrangement of two pulleys, as shown in the figure, is used to lift a 49.6-kg mass a distance of 3.70 m above the starting point. Assume the pulleys and rope are ideal and that all rope sections are essentially vertical.
(a) What is the mechanical advantage of this system? (In other words, by what factor is the force you exert to lift the weight multiplied by the pulley system?)
(b) What is the change in the potential energy of the weight when it is lifted a distance of 3.70 m?
kJ
(c) How much work must be done to lift the 49.6-kg mass a distance of 3.70 m?
kJ
(d) What length of rope must be pulled by the person lifting the weight 3.70 m higher in the air?
m
To answer these questions, let's break them down step by step:
Step 1: Analyze the pulley system
In this case, we have two pulleys arranged in a system. The mechanical advantage of a pulley system is equal to the number of supporting ropes. Since there are two supporting ropes in this system, the mechanical advantage is 2.
(a) The mechanical advantage is 2.
Step 2: Calculate the change in potential energy
The change in potential energy of an object can be calculated using the equation:
ΔPE = mgh
where
ΔPE is the change in potential energy
m is the mass of the object
g is the acceleration due to gravity
h is the height or distance moved vertically
Given:
m = 49.6 kg
h = 3.70 m
g = 9.8 m/s^2
(b) Let's substitute the given values into the formula:
ΔPE = (49.6 kg)(9.8 m/s^2)(3.70 m)
= 1819.488 J
The change in potential energy is 1819.488 J.
Step 3: Calculate the work done
The work done in lifting an object can be calculated using the equation:
W = Fd
where
W is the work done
F is the force applied
d is the distance moved in the direction of the force
Given:
m = 49.6 kg
h = 3.70 m
g = 9.8 m/s^2
(c) Since we are lifting the mass vertically, the force required is equal to the weight of the object, which can be calculated using the equation:
F = mg
F = (49.6 kg)(9.8 m/s^2)
= 485.28 N (approximately)
Let's substitute the values into the formula for work done:
W = (485.28 N)(3.70 m)
= 1797.696 J
The work done to lift the 49.6 kg mass a distance of 3.70 m is 1797.696 J.
Step 4: Calculate the length of rope pulled
Since the two pulleys essentially double the distance the rope is pulled, the length of rope pulled by the person lifting the weight 3.70 m higher in the air is half of the distance lifted.
(d) The length of rope pulled is 3.70 m/2 = 1.85 m.
Therefore, the person needs to pull 1.85 m of rope.
To answer these questions, we need to understand the concept of mechanical advantage, potential energy, work, and the relationships between them.
(a) Mechanical Advantage:
The mechanical advantage of a pulley system is the ratio of the output force (the force exerted to lift the weight) to the input force (the force applied by the person). In this case, since the pulleys and rope are ideal, the mechanical advantage is equal to the number of supporting strands in the rope. The figure shows that there are four supporting strands, so the mechanical advantage is 4.
(b) Change in Potential Energy:
The change in potential energy of an object lifted to a certain height is given by the formula: ΔPE = mgh, where ΔPE is the change in potential energy, m is the mass of the object, g is the acceleration due to gravity, and h is the height difference. Substituting the given values, we have:
ΔPE = (49.6 kg) * (9.8 m/s^2) * (3.70 m) = 1808.96 J
(c) Work Done:
The work done in lifting a mass is calculated using the formula: W = F * d, where W is the work done, F is the force applied, and d is the distance moved. In this case, the force applied is equal to the weight of the object (mg), so we have:
W = (49.6 kg) * (9.8 m/s^2) * (3.70 m) = 1808.96 J
(d) Length of Rope:
To determine the length of rope that needs to be pulled, we can double the distance lifted by the weight (3.70 m), since there are two strands of rope involved. Therefore, the length of rope pulled will be 2 * 3.70 m = 7.40 m.
So, to summarize:
(a) The mechanical advantage of the system is 4.
(b) The change in potential energy is 1808.96 J.
(c) The work done to lift the mass is 1808.96 J.
(d) The length of rope pulled is 7.40 m.