Mechanical Advantage The pulley system shown in the figure is used to lift a m= 67-kg crate. Note that one chain connects the upper pulley to the ceiling and a second chain connects the lower pulley to the crate. Assume the masses of the chains, pulleys, and ropes are negligible.
a) Determine the force F required to lift the crate with constant speed.
b) Determine the tension in the upper chain.
c) Determine the tension in the lower chain.
To solve this problem, we can use the principles of equilibrium and mechanical advantage. Let's define some variables:
m = mass of the crate = 67 kg
g = acceleration due to gravity = 9.8 m/s^2 (approximate value)
Now let's proceed with the solution:
a) Determine the force F required to lift the crate with constant speed:
To lift the crate with constant speed, the force F must balance the gravitational force acting on the crate. The gravitational force is given by:
F_grav = m * g
Substituting the values:
F_grav = 67 kg * 9.8 m/s^2
F_grav = 657.6 N
Therefore, the force F required to lift the crate with constant speed is 657.6 N.
b) Determine the tension in the upper chain:
In a pulley system, the tension in the chain is equal throughout the chain. Let's denote the tension in both the upper and lower chains as T.
Since the upper and lower chains connect to the crate separately, they each support half of the total weight of the crate.
Therefore, the tension in the upper chain (T_upper) can be calculated as:
T_upper = (1/2) * F_grav
T_upper = (1/2) * 657.6 N
T_upper = 328.8 N
Therefore, the tension in the upper chain is 328.8 N.
c) Determine the tension in the lower chain:
The tension in the lower chain (T_lower) is the same as the tension in the upper chain because they are connected and support the same load.
Therefore, the tension in the lower chain is also 328.8 N.
To solve this problem, we need to understand the concept of mechanical advantage in a pulley system. The mechanical advantage of a pulley system is defined as the ratio of the output force to the input force. In this case, the output force refers to the force applied to lift the crate, and the input force is the force applied to the chain or rope.
Let's define some variables to make it easier to solve the problem:
- F: Force required to lift the crate with constant speed.
- T_upper: Tension in the upper chain.
- T_lower: Tension in the lower chain.
- m: Mass of the crate.
- g: Acceleration due to gravity (approximately 9.8 m/s²).
a) Determine the force F required to lift the crate with constant speed:
In order to lift the crate with constant speed, the force F should be equal to the weight of the crate (mg). Therefore:
F = m * g
F = 67 kg * 9.8 m/s²
F ≈ 657.6 N
b) Determine the tension in the upper chain:
The tension in the upper chain is equal to the force required to lift the crate (F). Therefore:
T_upper = F
T_upper ≈ 657.6 N
c) Determine the tension in the lower chain:
In a pulley system, the tension remains constant throughout the entire system. So, the tension in the lower chain will also be equal to the force required to lift the crate (F).
T_lower = F
T_lower ≈ 657.6 N
Therefore, the tension in both the upper and lower chain is approximately 657.6 N.