Problem 7 - Applied Physics

Question

The MA of a hydraulic jack is \(250 .\) What force must be applied to lift an automobile weighing \(12,000 \mathrm{~N} ?\)

Step-by-Step Solution

Answer

The applied force must be 48 N.

Step 1: Understanding Mechanical Advantage

Mechanical Advantage (MA) is defined as the ratio of the output force (the force exerted by the machine) to the input force (the force applied to the machine). Mathematically, it is expressed as \( MA = \frac{F_{out}}{F_{in}} \), where \( F_{out} \) is the output force and \( F_{in} \) is the input force.

Step 2: Identifying Given Values

From the exercise, we know that the Mechanical Advantage \( MA \) is 250, and the output force \( F_{out} \), which is the weight of the automobile, is \( 12,000 \) N.

Step 3: Setting Up the Equation

We use the formula for Mechanical Advantage: \( MA = \frac{F_{out}}{F_{in}} \). We need to find \( F_{in} \), the input force. Rearranging the equation gives \( F_{in} = \frac{F_{out}}{MA} \).

Step 4: Calculating the Input Force

Substitute the known values into the rearranged formula: \[ F_{in} = \frac{12,000}{250} \].

Step 5: Performing the Calculation

Calculate \( F_{in} \) using the values: \( F_{in} = \frac{12,000}{250} = 48 \).

Step 6: Conclusion

The force that must be applied to the hydraulic jack to lift the automobile is \( 48 \) N.

Key Concepts

Understanding Hydraulic Jacks

Hydraulic jacks are crucial tools used to lift heavy loads easily. They utilize principles of fluid mechanics and pressure to multiply a small input force into a much larger output force. This is achieved through Pascal's principle, which states that when pressure is applied to a confined fluid, it is transmitted undiminished throughout the fluid. A hydraulic jack typically involves two cylinders: a small one where the input force is applied and a larger one responsible for lifting the load. By applying force to the smaller cylinder, it creates pressure in the fluid. This pressure results in a larger force on the larger cylinder, thereby lifting heavy objects with much less effort.

• Hydraulic jacks operate on fluid pressure
• Use Pascal's principle for force multiplication
• Consist of small input and larger output cylinders
• Ideal for lifting heavy loads with minimal force

When using hydraulic jacks, safety is vital. Always ensure the jack is on a firm, stable surface and never get under a lifted load without safety supports in place.

Calculating Input Force in Hydraulic Jacks

The input force is the force you need to apply to use a hydraulic jack effectively. It's determined by the concept of Mechanical Advantage (MA), which helps in calculating how much easier a task becomes due to the jack. The formula for MA is: \( MA = \frac{F_{out}}{F_{in}} \). In this formula, \( F_{out} \) represents the output force (force exerted by the jack) and \( F_{in} \) represents the input force (force applied by you).

To calculate the input force required to lift a heavy object using a hydraulic jack, rearrange the formula to solve for \( F_{in} \):

• Given \( MA \) = √250
• \( F_{out} \) (weight of the object) = 12,000 N
• Apply the formula: \( F_{in} = \frac{F_{out}}{MA} \)

Substituting the known values, you find:

• \( F_{in} = \frac{12,000}{250} = 48 \) N

This calculation shows the small force you need to apply to lift a huge weight using the jack's mechanical advantage.

Understanding Output Force

Output force in the context of hydraulic jacks is the force exerted by the jack to lift an object. It is a crucial factor in determining the effectiveness of a jack. The higher the output force, the greater the load the jack can lift. For example, in our exercise, the output force needed is 12,000 N to lift a car.
Output force is determined by the pressure created in the hydraulic system and the area over which this pressure acts. In a hydraulic jack:

• The output force depends on the cross-sectional area of the larger cylinder.
• Pressure is transmitted undiminished from the smaller cylinder to the larger cylinder.
• Output force is proportional to the size difference between the two cylinders.

This force and the mechanical advantage allow us to move substantial weights with relatively little effort. Understanding this principle helps in designing and using hydraulic systems more efficiently.