Relate work to change in kinetic energy
Scenario
Jab aap chalte hue skateboard ko piche se aur dhakka maarte ho, toh uski speed achanak se badh jaati hai.
Aapka lagaya hua 'work' actually skateboard mein kis form mein convert ho gaya?
Mental model
Jitna work aap karoge, object ki Kinetic Energy utni hi badh (ya ghat) jayegi. Work = change in Energy.
Ise Work-Energy Theorem kehte hain. Agar aap ek ghoomte pahiye par extra force laga kar work karte ho, toh wo saara work pahiye ki kinetic energy badhane mein chala jayega.
Explanation
Humne pehle padha tha ki work aur energy closely related hain. Work-energy theorem is rishte ko precisely mathematically define karti hai. Is theorem ke hisaab se, kisi object par net force laga kar jo work done hota hai, wo us object ki kinetic energy mein hone wale change ke barabar hota hai.
Socho ek m mass ka object initial velocity `u` se chal raha hai. Aapne uspar constant force `F` lagaya aur uski velocity badh kar `v` ho gayi. Jo work `W` aapne kiya, usne hi toh energy badhayi na?
Mathematically: `Work done = Final Kinetic Energy - Initial Kinetic Energy`.
Yani, `W = 1/2 m v^2 - 1/2 m u^2`, jise aap `W = 1/2 m (v^2 - u^2)` bhi likh sakte ho.
Ye bahut hi powerful tool hai kyunki aapko object ka acceleration ya distance nahi nikalna padta. Agar aapko initial aur final speed pata hai, toh aap directly nikal sakte ho ki force ne kitna work kiya hoga. Aur agar break lagane par speed kam ho rahi hai, toh final KE initial se kam hogi, jiska matlab change in kinetic energy negative aayega. Aur ye sense bhi banata hai kyunki force opposite direction mein tha (negative work)!
Worked example
Problem: What is the work to be done to increase the velocity of a car from 30 km/h to 60 km/h if the mass of the car is 1500 kg?
- Sabse pehle speed ko km/h se m/s mein convert karo (multiply by 5/18).
- Ab initial kinetic energy nikal lo.
- Fir final kinetic energy nikal lo.
- Work done nikalne ke liye dono ka difference le lo.
Answer: 156250 J
Problem: A 10 kg ball initially at rest is pushed so that it reaches a speed of 2 m/s. Find the work done on the ball.
- Object start mein rest par hai toh initial velocity zero hai.
- Initial KE zero hogi. Work done equals final KE.
- Solve kar do.
Answer: 20 J
Key points
- Work-Energy Theorem: Work done on an object is equal to the change in its kinetic energy.
- Formula: W = 1/2 m (v^2 - u^2)
Common mistakes
- Subtracting speeds first: Kuch students (v - u) nikal kar uska square kar dete hain, yani 1/2 m (v - u)^2. Ye galat hai! Aapko pehle dono ke squares nikalne hain, fir unko subtract karna hai: (v^2 - u^2).
Recall questions
- What does the work-energy theorem state?
- If the kinetic energy of an object decreases, what can you say about the work done on it?
- What is the final kinetic energy if a moving object is brought to a complete stop?
Questions & answers
Certain force acting on a 20 kg mass changes its velocity from 5 m/s to 2 m/s. Calculate the work done by the force.
-210 J
Approach: Initial KE = 0.5 * 20 * 25 = 250 J. Final KE = 0.5 * 20 * 4 = 40 J. Work done = 40 - 250 = -210 J. The negative sign shows force acts opposite to motion.
A force acts on a 2 kg object so that its velocity increases from 2 m/s to 3 m/s. What is the work done?
5 J.
Approach: W = 0.5 * 2 * (3^2 - 2^2) = 1 * (9 - 4) = 5 J.
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