Concepts

Work

• Definition: Work is done when a force causes an object to move in the direction of the force. Mathematically, work is calculated as the dot product of the force and the displacement.

• Positive and negative work: Work is done by a constant force in the direction of the force is taken to be positive and against the direction of the force as negative.

• Units of work: The SI unit of work is the joule (J), which is defined as the work done when a force of one newton (N) is applied over a distance of one meter (m) in the direction of the force.

Energy:

• Definition: Energy is the ability to do work or cause change. It is a scalar quantity.

• Various forms of energy: There are many different forms of energy, including kinetic energy (the energy of motion), potential energy (the energy stored in a system due to its position or configuration), thermal energy (the energy associated with the random motion of particles), and electrical energy (the energy associated with the movement of electric charges).

• Units of energy: The SI unit of energy is the joule (J), which is the same unit used to measure work.

Power:

• Definition: Power is the rate at which work is done or energy is transferred. Mathematically, power is calculated as the scalar product of the force and the velocity of the object.

• Scalar quantity: Power is a scalar quantity, meaning it has only magnitude and no direction.

• Relationship between work, energy, and power: Power is related to work and energy by the equation:

Power = Work / Time 
Energy = Work done = Power x Time

• Units of power: The SI unit of power is the watt (W), which is defined as one joule per second (1 W = 1 J/s).

Example Problems

Calculating the work done by a force acting on an object over a distance

A force of 10 N is applied to an object, causing the object to move a distance of 5 meters in the direction of the force. Calculate the work done.

Solution:

Using the definition of work, we have:

Work = Force x Displacement x cosθ
 Work = 10 N x 5 m x cos 0°
 Work = 50 J

Therefore, the work done by the force is 50 J.

Determining the kinetic energy of an object based on its mass and velocity

An object with a mass of 10 kg is moving at a velocity of 5 m/s. Calculate the kinetic energy of the object.

Solution:

Using the formula for kinetic energy, we have:

 Kinetic energy = (1 / 2)  mv^2
 Kinetic energy = (1 / 2) x 10 kg x (5 m/s)^2
 Kinetic energy = 125 J

Therefore, the kinetic energy of the object is 125 J.

Calculating the potential energy of an object based on its mass, height, and acceleration due to gravity

An object with a mass of 20 kg is 10 meters above the ground. Calculate the potential energy of the object due to its position in the Earth’s gravitational field (acceleration due to gravity = 9.8 m/s2).

Solution:

Using the formula for potential energy, we have:

Potential energy = mgh
Potential energy = 20 kg x 9.8 m/s2 x 10 m
Potential energy = 1960 J

Therefore, the potential energy of the object due to its position is 1960 J.

Calculating the power required to lift an object at a certain speed against the force of gravity

A force of 500 N is required to lift an object a height of 10 meters in 2 seconds. Calculate the power required to perform this task against gravity.

Solution:

Using the definition of power, we have:

Power =  Work done / Time

First, calculate the work done by the force:

Work done = Force x Displacement x cos θ
Work done = 500 N x 10 m x cos 0° = 5000 J

Then, substitute work and time into the power equation:

Power = Work done / Time
Power = 5000 J / 2 s = 2500 W

Therefore, the power required to lift the object is 2500 W.