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Energy and power are fundamental concepts in physics that explain how things move and how work is done in the world around us. Whether it is lifting water from a well, moving a vehicle, or powering household appliances, understanding energy and power helps us grasp how these actions occur and how efficiently they are performed.
Energy is the capacity to do work. It exists in many forms, such as the energy of motion or the energy stored in an object due to its position. Power is the rate at which work is done or energy is transferred from one form to another.
These concepts are measured using metric units. Energy is measured in joules (J), and power is measured in watts (W), where 1 watt equals 1 joule per second. This system allows us to compare energy use and power output accurately across various machines and processes.
In competitive exams and real-life engineering problems, mastering energy and power concepts enables you to solve practical questions about motion, machines, electrical devices, and more.
Energy comes in various forms, but two important types are kinetic energy and potential energy. Mechanical energy is the sum of these two.
Kinetic energy is the energy possessed by a body due to its motion. For example, when a car moves on the road or a cricket ball is thrown, they have kinetic energy.
Potential energy is the energy stored in an object because of its position or configuration. For example, a stretched bow, a raised weight on a shelf, or water stored at a height in a dam all have potential energy.
When considering motion and position together, the total mechanical energy of a system is the sum of kinetic and potential energy. This combined energy determines how objects move and interact.
Explanation: This diagram shows a pendulum bob swinging. At the highest point (left), the bob has maximum potential energy due to its height and almost zero kinetic energy because it's momentarily at rest. At the lowest point (right), the bob has maximum kinetic energy as it moves fastest and minimum potential energy since it is at the lowest height. This illustrates how energy continuously shifts from potential to kinetic and back during the swing.
Work in physics is done when a force causes an object to move in the direction of the force. Specifically, work done is the product of the force applied, the displacement it causes, and the cosine of the angle between the force and displacement directions.
Mathematically, work is defined as
\[ W = F d \cos \theta \]where:
Work can be positive, negative, or zero based on the angle between the force and displacement:
graph TD A[Force and Displacement] --> B1[Angle < 90°] A --> B2[Angle = 90°] A --> B3[Angle > 90°] B1 --> C1[Work is Positive] B2 --> C2[Work is Zero] B3 --> C3[Work is Negative]
Examples:
Sometimes, the force is not constant but changes with position. In such cases, work done is calculated by integrating the force over the displacement:
\[ W = \int F(x) \, dx \]This means you add up tiny amounts of work done through very small steps across the displacement.
Power is the rate at which work is done or energy is transferred. It tells us how quickly energy moves from one form to another or how fast work is done.
The formula for power is:
\[ P = \frac{W}{t} \]Where:
For example, a motor rated at 1500 W uses 1500 joules of energy every second it operates. When calculating electricity costs, this helps estimate how much energy appliances consume and their cost over time.
The Law of Conservation of Energy states that energy cannot be created or destroyed; it can only change from one form to another. In a closed system without friction or external forces, the total mechanical energy remains constant.
Consider the example of a pendulum swinging without air resistance:
The pendulum transforms potential energy to kinetic energy as it swings down, and kinetic energy back to potential energy as it swings up. This cycle continues, demonstrating energy conservation.
Step 1: Convert velocity from km/h to m/s.
\( 72 \; \text{km/h} = \frac{72 \times 1000}{3600} = 20 \; \text{m/s} \)
Step 2: Use formula for kinetic energy, \( KE = \frac{1}{2} m v^2 \).
\( KE = \frac{1}{2} \times 1500 \times (20)^2 = 750 \times 400 = 300,000 \; \text{J} \)
Answer: The kinetic energy of the car is \( 3.0 \times 10^{5} \) joules (J).
Step 1: Identify values: \( F = 50 \, \text{N}, \; d = 5 \, \text{m}, \; \theta = 30^{\circ} \).
Step 2: Use work formula: \( W = F d \cos \theta \).
\( W = 50 \times 5 \times \cos 30^{\circ} = 250 \times 0.866 = 216.5 \; \text{J} \).
Answer: The work done by the force is approximately 216.5 joules.
Step 1: Calculate energy consumed in kilowatt-hours (kWh):
Power = 1500 W = 1.5 kW
Time = 3 hours
Energy = Power x Time = 1.5 x 3 = 4.5 kWh
Step 2: Calculate cost:
Cost = Energy x Rate = 4.5 x Rs.7 = Rs.31.5
Answer: The motor consumes 4.5 kWh energy and the cost is Rs.31.50.
Step 1: At the highest point, the bob has potential energy and zero kinetic energy. At the lowest point, potential energy is zero and kinetic energy is maximum.
Step 2: Use the conservation of mechanical energy:
\( PE_{\text{top}} = KE_{\text{bottom}} \)
\( m g h = \frac{1}{2} m v^2 \)
Simplifying, \( v = \sqrt{2 g h} \)
Step 3: Substitute values \( g = 9.8 \, m/s^2, h = 0.5 \, m \):
\( v = \sqrt{2 \times 9.8 \times 0.5} = \sqrt{9.8} \approx 3.13 \, m/s \)
Answer: The speed of the pendulum bob at the lowest point is approximately 3.13 m/s.
Step 1: Use formula for work done by variable force:
\( W = \int_{0}^{4} 5x \, dx \)
Step 2: Calculate integral:
\( W = 5 \int_{0}^{4} x \, dx = 5 \left[ \frac{x^2}{2} \right]_{0}^{4} = 5 \times \frac{4^2}{2} = 5 \times \frac{16}{2} = 5 \times 8 = 40 \, \text{J} \)
Answer: The work done by the variable force over 4 meters is 40 joules.
When to use: Whenever velocity is given in km/h or other units.
When to use: When force is applied at an angle to displacement.
When to use: Estimating electricity consumption for household appliances.
When to use: Problems with pendulums, roller coasters, or inclined planes.
When to use: When force varies with position.
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