Work done by a constant and variable force

In physics, the concepts of work, energy, and power are central to understanding how forces affect the motion of objects. Within this framework, the idea of work done by a force plays a key role. We will take a deeper look at how work is defined and calculated when dealing with both constant and variable forces.

What is the work?

In everyday language, "work" can include any physical or mental effort. However, in physics, work has a precise definition. It is the product of the force applied to an object and the displacement in the direction of the force.

Formula of work

The basic equation to calculate the work done by a force is:

Work (W) = Force (F) × Displacement (d) × cos(θ)

Where:

• W is the work done
• F is the magnitude of the applied force
• d is the displacement caused by the force
• θ is the angle between the direction of force and displacement

Work done by a constant force

Constant force means that the force applied remains unchanged in magnitude and direction over time. Calculating the work done by a constant force is straightforward because of its simplicity.

Example: pulling a sled

Suppose you are pulling a sled on a horizontal surface by applying a force of 50 N. The sled moves 10 m while you are pulling it, and the force is applied in the direction of motion. In this case the angle is 0 degrees (because the force is in the same direction as the displacement).

The work done may be calculated as follows:

Work = 50 N × 10 m × cos(0°) = 500 N·m (or Joules)

Visual example:

Work done by a variable force

A variable force changes in either magnitude or direction (or both) during the displacement. Calculating the work done becomes more complicated because you can't use simple multiplication of variables.

Linear force example:

Imagine a spring being compressed. The force exerted by the spring is described by Hooke's law, which says:

F = -kx

where k is the spring constant and x is the displacement from the equilibrium position.

Calculation of work done

The work done by a variable force is calculated using integration. The work done in moving an object from position a to b is given by:

W = ∫ from a to b F(x) dx

Example of work done by spring

When a spring with spring constant k is compressed by a distance x, the work done is calculated as:

W = ∫ from 0 to x (-kx) dx = -[1/2 kx^2] from 0 to x = -1/2 kx^2

Because we only care about the magnitude of the work in most practical scenarios, the work done on the spring is 1/2 kx^2.

Visual example:

Important considerations in the work

When calculating work, several factors must be considered in order to correctly interpret the scenario:

• If the displacement is zero, no work is done, even if force is applied.
• If the force is perpendicular to the direction of displacement, no work is done (cos(90°) = 0).
• When the force opposes the motion, the work done is negative, which indicates that energy is removed from the system.

Example: Lifting an object vertically

Suppose an object weighing 10 kg is lifted 2 m upwards. The force required is equal to the weight of the object, as follows:

F = mass × gravity = 10 kg × 9.8 m/s^2 = 98 N

The work done is:

W = 98 N × 2 m × cos(0°) = 196 N·m (or Joules)

Visual example:

Applications and examples

Understanding the work done by forces is important in many real-life situations, from simple machines to complex mechanical systems. This concept is applied as follows:

Pulley systems

Lifting a heavy object can be made easier with a pulley system. The amount of work needed to lift an object is the same, but by using multiple pulleys, the force required can be distributed over a greater distance; therefore, less force is needed.

Suppose you need to lift a 200 kg crate onto a platform 5 m high. Without the pulley, the force required would be:

F = mass × gravity = 200 kg × 9.8 m/s^2 = 1960 NW = F × d = 1960 N × 5 m = 9800 J

Vehicle acceleration

Cars accelerate by using the work done by the engine against resistive forces such as friction. If a car's engine exerts force to overcome friction and move the car, this can be calculated using variable force concepts to figure out how energy is consumed.

Wind turbines

Wind turbines convert kinetic energy from the wind into mechanical energy using the work done on the turbine blades by the wind force. Here, the work calculation involves an integration approach because the wind force varies.

Conclusion

Work done by forces in physics is a concept that connects the abstract with the practical. Whether dealing with static forces such as those exerted by a car engine or variable forces such as the compression of a spring or the generation of electricity through wind, principles remain fundamental to the effective understanding and application of mechanical systems. By understanding both the calculations and the applications, we gain valuable insights into the principles that govern momentum and energy interactions.

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