Overview
This lecture covers the work energy theorem in detail, including its definition, mathematical formulas, and real-life applications. It also explains how work, kinetic energy, and potential energy are related, discusses the effects of resistive forces, and explores energy losses and efficiency in machines.
Work Energy Theorem
- The work energy theorem states that the net work done on an object is equal to the change in its kinetic energy.
- When a force is applied to move an object, the object travels a certain distance, and the product of force and displacement is the work done.
- As soon as work is done on an object, its kinetic energy changes because its speed changes.
- In simple terms, the work done on an object equals the change in its kinetic energy.
- Mathematically:
- Work done = Change in Kinetic Energy
- Work done = ΔK.E. = ½mv² – ½mu²
- m = mass of the object
- u = initial velocity
- v = final velocity
- This means that when a force causes an object to accelerate, the work done by the force changes the object's kinetic energy.
- Example: If a car accelerates from rest to 10 m/s, the work done by the engine increases the car's kinetic energy.
Work Energy Theorem and Potential Energy
- The work energy theorem also applies to potential energy, especially for conservative forces like gravity.
- When work is done against gravity (such as lifting a box), the work done is stored as potential energy in the object.
- Mathematically:
- Work done = Change in Potential Energy
- Work done = ΔP.E. = Final Potential Energy – Initial Potential Energy
- As an object is lifted, its potential energy increases. When released, this potential energy converts back into kinetic energy as the object falls.
- Potential and kinetic energy change in opposite directions: as one increases, the other decreases, which is why a negative sign is used when relating their changes.
- In a conservative system, work done by forces like gravity is stored as potential energy, which can later convert back to kinetic energy.
Work Energy Theorem in Resistive Medium
- In real situations, objects often move through resistive media such as air or surfaces with friction.
- When a force is applied to move an object in a resistive medium, part of the work done increases the object's kinetic energy, while another part is used to overcome resistive forces (like friction or air resistance).
- The direction of the resistive force is always opposite to the direction of the applied force, reducing the net work done on the object.
- As a result, not all the work done by the applied force is converted into kinetic energy; some energy is lost due to resistance.
- This energy loss slows down the object or prevents it from accelerating further.
Energy Losses in Machines
- Most practical machines, such as pulleys and engines, do not convert all input energy into useful work.
- A significant portion of the input energy is lost, mainly as heat and friction.
- Input energy is calculated as:
- Input = Effort × Distance (P × d), where P is the applied force (effort) and d is the distance moved.
- Output energy is the useful work done:
- Output = Weight × Distance (W × d), where W is the load lifted and d is the distance it is lifted.
- In real devices, only a part of the input energy is converted into useful output; the rest is wasted.
Ideal vs Real Machine
- An ideal machine is a theoretical concept where 100% of the input energy is converted into useful work, with no energy lost.
- In reality, all machines lose some energy due to factors like friction and heat, so their efficiency is always less than 100%.
- For example, in a pulley system, if 60% of the energy is lost due to friction, only 40% is available for lifting the load.
- The efficiency of a machine depends on how much of the input energy is converted into useful output.
Efficiency of Machine
- Efficiency measures how well a machine converts input energy into useful output work.
- Formula:
- Efficiency = (Useful Output Work / Input Work) × 100%
- The higher the efficiency, the better the machine performs.
- Example: If an engine receives 1000 joules of chemical energy but only does 100 joules of useful work, its efficiency is (100/1000) × 100% = 10%. The remaining 900 joules are lost as heat or friction.
- In real engines (like those in cars or bikes), efficiency typically ranges from 35% to 50%, with the rest of the energy lost.
Key Terms & Definitions
- Work Done: The energy transferred when a force moves an object over a distance.
- Kinetic Energy: The energy an object has due to its motion, calculated as ½mv².
- Potential Energy: The energy stored in an object due to its position, especially in a gravitational field.
- Resistive Force: A force that opposes motion, such as friction or air resistance.
- Efficiency: The ratio of useful output work to input energy, expressed as a percentage.
- Ideal Machine: A hypothetical machine with no energy loss and 100% efficiency.
Action Items / Next Steps
- Watch the remaining videos in the chapter playlist for further topics.
- Practice solving problems related to machine efficiency and the work energy theorem.
- Review solved assignments, numericals, and short questions available in the provided resources.