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
a) The mechanical advantage of this system is 2.
b) The change in the potential energy of the weight when it is lifted a distance of 3.70 m is 472.32 kJ.
c) The work done to lift the 49.6-kg mass a distance of 3.70 m is 472.32 kJ.
d) The length of rope that must be pulled by the person lifting the weight 3.70 m higher in the air is 7.40 m.
To answer these questions, we need to use the principles of work, energy, and mechanical advantage. Here's how you can calculate each part of the problem:
(a) To find the mechanical advantage of the system, we need to know the force exerted by the person lifting the weight and the force required to support the weight. Since the pulleys are ideal, the force required to support the weight is equal to the weight itself. The weight of the mass is given as 49.6 kg.
The mechanical advantage of a pulley system is given by the formula:
Mechanical Advantage = (Force input) / (Force output)
In this case, the force input is the force exerted by the person (F input), and the force output is the weight of the mass (F output). So, the mechanical advantage can be calculated as:
Mechanical Advantage = F input / F output = F input / (49.6 kg * g), where g is the acceleration due to gravity (9.8 m/s^2).
(b) The potential energy of an object is given by the formula:
Potential Energy = m * g * h
where m is the mass of the object, g is the acceleration due to gravity, and h is the height or distance above the starting point. In this case, the mass is given as 49.6 kg, and the height is given as 3.70 m. So, the change in potential energy can be calculated as:
Change in Potential Energy = m * g * h
(c) The work done to lift an object is equal to the force applied multiplied by the distance over which the force is exerted. In this case, the force applied is the weight of the mass (49.6 kg * g) and the distance is given as 3.70 m. So, the work done can be calculated as:
Work = Force * Distance = (49.6 kg * g) * 3.70 m
(d) The length of the rope that needs to be pulled by the person lifting the weight can be calculated using the formula:
Length of rope = 2 * distance
Since the rope sections are essentially vertical, the length of the rope is equal to twice the distance over which the weight is lifted. In this case, the distance is given as 3.70 m. So, the length of the rope can be calculated as:
Length of rope = 2 * 3.70 m
Now, you can use the given values and equations to find the answers to parts (a), (b), (c), and (d) of the problem.
To solve this problem, we need to consider the principles of work, energy, and mechanical advantage.
(a) The mechanical advantage of a pulley system can be calculated by counting the number of ropes supporting the load. In this case, there are two segments of rope supporting the load. Therefore, the mechanical advantage is 2.
(b) The change in potential energy can be calculated using the formula:
ΔPE = m * g * h
where m is the mass, g is the acceleration due to gravity (9.8 m/s^2), and h is the height or distance lifted.
ΔPE = 49.6 kg * 9.8 m/s^2 * 3.70 m = 1805.04 kg·m^2/s^2
To convert this to kilojoules (kJ), we divide by 1000:
ΔPE = 1805.04 kg·m^2/s^2 / 1000 = 1.80504 kJ
Therefore, the change in potential energy of the weight is approximately 1.805 kJ.
(c) The work done to lift an object is equal to the change in potential energy, which we calculated in part (b).
Work = ΔPE = 1.805 kJ
Therefore, the work required to lift the 49.6-kg mass a distance of 3.70 m is approximately 1.805 kJ.
(d) The length of rope that needs to be pulled can be calculated using the formula for the circumference (C) of a circle:
C = 2 * π * r
where r is the radius of the pulley. Since the pulley system is arranged such that all rope sections are essentially vertical, the length of rope that needs to be pulled is equal to twice the height (2 * h) lifted by the weight.
Length of rope = 2 * 3.70 m = 7.40 m
Therefore, the length of rope that must be pulled by the person lifting the weight 3.70 m higher in the air is approximately 7.40 m.