Notes : Derivation of Hydraulic Lift Equation - Class 11 Physics - JEE NEET

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Learn Hydraulic Machines Class 11 Physics with complete NCERT notes, Pascal's Law, hydraulic lift derivation, formulas, MCQs, FAQs, and important questions - Physicskund 

Hydraulic Machines

Hydraulic machines are devices that work on Pascal's Law. They use a confined liquid to transmit pressure from one point to another. Common examples include hydraulic lifts, hydraulic brakes, hydraulic jacks, and hydraulic presses.

Pascal's Law

Statement:

Whenever external pressure is applied on any part of a confined fluid, it is transmitted equally and undiminished in all directions.

This principle forms the basis of all hydraulic machines.

Demonstration of Pascal's Law

Consider a horizontal cylinder completely filled with a liquid and connected to three vertical tubes of different shapes.

Hydraulic Machines

Before Applying Pressure

  • The liquid stands at the same level in all three tubes.
  • This shows that pressure at the same horizontal level is the same throughout the liquid.

After Applying Pressure

  • When the piston is pushed, pressure is applied to the confined liquid.
  • The applied pressure is transmitted equally throughout the liquid.
  • The liquid rises in all three tubes.
  • Finally, the liquid reaches the same level in every tube, although the tubes have different shapes.

Conclusion: External pressure applied to a confined liquid is transmitted equally and without loss in all directions. This verifies Pascal's Law.

Hydraulic Lift

A hydraulic lift is an important application of Pascal's Law. It is used to lift heavy loads, such as cars and trucks, by applying a comparatively small force.

Hydraulic Machines

A hydraulic lift consists of:

  • A small piston of cross-sectional area A1.
  • A large piston of cross-sectional area A2.
  • A confined liquid between the two pistons.

A force F1 is applied to the small piston, while the load is placed on the large piston.

Derivation of Hydraulic Lift Equation

Step 1: Pressure on the Small Piston

When a force F1 is applied on the small piston, the pressure produced is

$$P=\frac{F_1}{A_1}$$

where,

  • P = Pressure
  • F1 = Applied force
  • A1 = Area of the small piston

Step 2: Transmission of Pressure

According to Pascal's Law, the pressure is transmitted equally throughout the liquid.

Therefore, the pressure acting on the large piston is

$$P=\frac{F_2}{A_2}$$

Step 3: Equating the Pressures

Since the pressure is the same throughout the liquid,

$$\frac{F_1}{A_1}=\frac{F_2}{A_2}$$

Multiplying both sides by A2,

$$F_2=\frac{F_1A_2}{A_1}$$

Hence,

$$\boxed{F_2=\frac{F_1A_2}{A_1}}$$

Working of Hydraulic Lift

  1. A small force F1 is applied to the small piston.
  2. This produces pressure in the confined liquid.
  3. According to Pascal's Law, the pressure is transmitted equally throughout the liquid.
  4. The same pressure acts on the large piston.
  5. Since the large piston has a greater area, a much larger upward force F2 is produced.
  6. Thus, heavy loads can be lifted with a comparatively small applied force.

Mechanical Advantage

Mechanical advantage is the ratio of output force to input force.

$$\text{Mechanical Advantage}=\frac{F_2}{F_1}$$

Substituting the value of F2,

$$\boxed{\text{Mechanical Advantage}=\frac{A_2}{A_1}}$$

A larger value of A2/A1 gives a greater lifting force.

Applications of Hydraulic Machines

  • Hydraulic lift
  • Hydraulic brakes
  • Hydraulic jack
  • Hydraulic press

Key Formulae

Pressure:

$$P=\frac{F}{A}$$

Pressure on the small piston:

$$P=\frac{F_1}{A_1}$$

Pressure on the large piston:

$$P=\frac{F_2}{A_2}$$

Hydraulic Lift Equation:

$$F_2=\frac{F_1A_2}{A_1}$$

Mechanical Advantage:

$$\text{Mechanical Advantage}=\frac{A_2}{A_1}$$

Key Points

  • Hydraulic machines work on Pascal's Law.
  • Pressure applied to a confined liquid is transmitted equally and undiminished in all directions.
  • Pressure remains the same throughout the liquid.
  • A larger piston produces a greater force because it has a larger area.
  • A small applied force can lift a heavy load.
  • The force multiplication depends on the ratio A2/A1.

Frequently Asked Questions (FAQs)

1. What is a hydraulic machine?

A hydraulic machine is a device that works on Pascal's Law and uses a confined liquid to transmit pressure from one point to another.

2. On which principle does a hydraulic lift work?

A hydraulic lift works on Pascal's Law.

3. Why is oil used in hydraulic machines?

Oil is nearly incompressible and transmits pressure efficiently throughout the system.

4. Why is the force on the large piston greater than the force on the small piston?

Because the large piston has a greater cross-sectional area. The same pressure acts over a larger area, producing a greater force.

5. What is the mechanical advantage of a hydraulic lift?

The mechanical advantage is given by:

$$\text{Mechanical Advantage}=\frac{A_2}{A_1}$$

Quiz (MCQs)

1. Hydraulic machines work on

  • A. Bernoulli's Principle
  • B. Pascal's Law
  • C. Archimedes' Principle
  • D. Hooke's Law

Answer: B. Pascal's Law

2. Pressure applied to a confined liquid is transmitted

  • A. Only upward
  • B. Only downward
  • C. Equally in all directions
  • D. Only sideways

Answer: C. Equally in all directions

3. The large piston of a hydraulic lift has

  • A. Smaller area
  • B. Equal area
  • C. Larger area
  • D. No area

Answer: C. Larger area

4. The SI unit of pressure is

  • A. Joule
  • B. Pascal
  • C. Newton
  • D. Watt

Answer: B. Pascal

5. Hydraulic brakes are an application of

  • A. Pascal's Law
  • B. Newton's Law
  • C. Bernoulli's Principle
  • D. Boyle's Law

Answer: A. Pascal's Law

True or False

  1. Hydraulic machines work on Pascal's Law. (True)
  2. Pressure is transmitted equally throughout a confined liquid. (True)
  3. The large piston has a smaller area than the small piston. (False)
  4. Hydraulic lifts can lift heavy loads using a small force. (True)
  5. Hydraulic brakes are based on Pascal's Law. (True)

Fill in the Blanks

  1. Hydraulic machines work on Pascal's Law.
  2. The pressure applied to a confined liquid is transmitted equally in all directions.
  3. The cross-sectional area of the small piston is represented by A1.
  4. The output force on the large piston is represented by F2.
  5. The SI unit of pressure is Pascal (Pa).

Very Short Answer Questions

1. What is Pascal's Law?

Pascal's Law states that pressure applied to a confined fluid is transmitted equally and undiminished in all directions.

2. Name two applications of hydraulic machines.

Hydraulic lift and hydraulic brakes.

3. Write the formula for pressure.

$$P=\frac{F}{A}$$

4. What does A1 represent?

A1 is the cross-sectional area of the small piston.

5. What does F2 represent?

F2 is the force acting on the large piston.

Short Answer Questions

1. Explain the working of a hydraulic lift.

When a force is applied on the small piston, pressure is produced in the confined liquid. According to Pascal's Law, this pressure is transmitted equally throughout the liquid. The same pressure acts on the larger piston and produces a greater force, enabling heavy loads to be lifted.

2. Why does a hydraulic lift multiply force?

Because the same pressure acts on a larger piston area, producing a greater force.

3. Write the expression for the mechanical advantage of a hydraulic lift.

$$\text{Mechanical Advantage}=\frac{F_2}{F_1}=\frac{A_2}{A_1}$$

Long Answer Questions

1. State Pascal's Law and derive the expression for the force produced in a hydraulic lift.

Pascal's Law: Pressure applied to a confined fluid is transmitted equally and undiminished in all directions.

Pressure on the small piston:

$$P=\frac{F_1}{A_1}$$

Pressure on the large piston:

$$P=\frac{F_2}{A_2}$$

Equating the pressures,

$$\frac{F_1}{A_1}=\frac{F_2}{A_2}$$

Therefore,

$$F_2=\frac{F_1A_2}{A_1}$$

This equation shows that a small force can lift a heavy load when the area of the large piston is greater than that of the small piston.

2. Derive the expression for the mechanical advantage of a hydraulic lift.

Mechanical advantage is the ratio of output force to input force.

$$\text{Mechanical Advantage}=\frac{F_2}{F_1}$$

Using

$$F_2=\frac{F_1A_2}{A_1}$$

we get

$$\boxed{\text{Mechanical Advantage}=\frac{A_2}{A_1}}$$

Thus, the mechanical advantage of a hydraulic lift is equal to the ratio of the cross-sectional areas of the two pistons.

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