Work-Energy Theorem, it relates to the variation of kinetic energy of an object due to the work done on it. The theorem states that the net work done on an object is equal to the change in its kinetic energy. Mathematically, it can be expressed as:
Work (W) = Change in Kinetic Energy (ΔKE)
The formula for calculating work is given by:
Work (W) = Force (F) * Distance (d) * cos(θ)
where θ is the angle between the force vector and the displacement vector.
Let's solve two simple problems using the Work-Energy Theorem:
Problem 1:
A block with a mass of 2 kg is initially at rest. A force of 10 N is applied horizontally to the block, causing it to move a distance of 5 meters. Calculate the work done and the resulting kinetic energy of the block.
Solution:
Given:
Mass (m) = 2 kg
Force (F) = 10 N
Distance (d) = 5 m
The work done can be calculated using the formula:
Work (W) = Force (F) * Distance (d) * cos(θ)
Since the force and displacement are in the same direction (horizontal), the angle θ is 0 degrees, and cos(0) = 1.
Work (W) = 10 N * 5 m * cos(0)
= 50 J (joules)
According to the Work-Energy Theorem, the work done is equal to the change in kinetic energy:
W = ΔKE
Therefore, the change in kinetic energy is 50 joules.
Since the block was initially at rest, its initial kinetic energy (KEi) is zero. Thus, the final kinetic energy (KEf) is:
KEf = KEi + ΔKE
= 0 + 50 J
= 50 J
Therefore, the resulting kinetic energy of the block is 50 joules.
Problem 2:
A car of mass 1,500 kg is initially moving with a velocity of 20 m/s. An applied force of 2,000 N acts in the direction of the car's motion for a distance of 100 meters. Determine the work done and the resulting change in kinetic energy.
Solution:
Given:
Mass (m) = 1,500 kg
Force (F) = 2,000 N
Distance (d) = 100 m
The work done can be calculated as:
Work (W) = Force (F) * Distance (d) * cos(θ)
Since the force and displacement are in the same direction (direction of motion), the angle θ is 0 degrees, and cos(0) = 1.
Work (W) = 2,000 N * 100 m * cos(0)
= 200,000 J (joules)
According to the Work-Energy Theorem, the work done is equal to the change in kinetic energy:
W = ΔKE
Therefore, the change in kinetic energy is 200,000 joules.
The initial kinetic energy (KEi) of the car is given by:
KEi = (1/2) * mass * velocity^2
= (1/2) * 1,500 kg * (20 m/s)^2
= 300,000 J (joules)
The final kinetic energy (KEf) is:
KEf = KEi + ΔKE
= 300,000 J + 200,000 J
= 500,000 J
Therefore, the resulting kinetic energy of the car is 500,000 joules.
In summary:
The work done on the car is 200,000 joules.
The initial kinetic energy of the car is 300,000 joules.
The change in kinetic energy is 200,000 joules.
The final kinetic energy of the car is 500,000 joules.
The Work-Energy Theorem states that the work done on an object is equal to the change in its kinetic energy. In both examples, we used this theorem to relate the work done on the object to its resulting change in kinetic energy.
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