the input and output of hydraulic jack are respectively 1 cm and 4 cm in diameter. A lever with a mechanical advantage of 6 is used to apply force to the input piston. How much mass can the jack lift if a force of 180 N is applied to the lever and efficiency is 80%
1410.6 kg
1410.6KG
student
To calculate the mass that the hydraulic jack can lift, we need to consider the mechanical advantage and efficiency of the system.
First, let's calculate the force applied to the output piston of the hydraulic jack using the mechanical advantage formula:
Mechanical Advantage = Output Force / Input Force
Given that the mechanical advantage is 6 and the force applied to the lever is 180 N, we can rearrange the equation to solve for the output force:
Output Force = Mechanical Advantage * Input Force
Output Force = 6 * 180 N
Output Force = 1080 N
Now, let's consider the efficiency of the system. Efficiency is defined as the ratio of output power to input power. In this case, power is the product of force and distance. However, since the distance is not given, we will assume that the distance traveled by the input piston is the same as the distance traveled by the output piston. Therefore, we can calculate the efficiency as follows:
Efficiency = (Output Force * Output Distance) / (Input Force * Input Distance)
Since the input and output diameters are given, we can use the formula for the area of a circle to calculate the corresponding distances:
Input Distance = (π * (Input Diameter / 2)^2) = (π * (1 cm / 2)^2) = (π * 0.25 cm^2)
Input Distance = 0.785 cm^2
Output Distance = (π * (Output Diameter / 2)^2) = (π * (4 cm / 2)^2) = (π * 4 cm^2)
Output Distance = 12.57 cm^2
Substituting the values into the efficiency formula:
0.80 = (1080 N * 12.57 cm^2) / (180 N * 0.785 cm^2)
Now, we can solve for the output force:
0.80 = (1080 N * 12.57 cm^2) / (180 N * 0.785 cm^2)
0.80 = 13,561.6 / 141.3
0.80 = 96.05
Since the efficiency value is incorrect, we need to recalculate the output force.
Output Force = (0.80 * Input Force * Input Distance) / Output Distance
Output Force = (0.80 * 180 N * 0.785 cm^2) / 12.57 cm^2
Output Force = 112.56 N
Finally, we can calculate the mass that the jack can lift using Newton's second law:
Force = Mass * Acceleration
Rearranging the equation to solve for mass:
Mass = Force / Acceleration
Considering the acceleration due to gravity (usually denoted as g = 9.8 m/s^2):
Mass = Output Force / g
Mass = 112.56 N / 9.8 m/s^2
Mass = 11.49 kg
Therefore, the hydraulic jack can lift a mass of approximately 11.49 kg if a force of 180 N is applied to the lever and the efficiency is 80%.