Work Done by a Constant and Variable Force

Introduction

In physics, work is a measure of energy transfer that occurs when an object is moved over a distance by an external force. It is a crucial concept for understanding how energy is used and transformed in mechanical systems. The calculation of work depends on whether the force applied is constant or variable.

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1. Work Done by a Constant Force

Definition

Work is done by a constant force when the force applied to an object does not change in magnitude or direction, and the object undergoes a displacement.

Formula:
The work done (W) is defined as the product of the magnitude of the displacement (d) and the component of the force (F) that is parallel to the displacement.
$$W = (F \cos\theta) d$$
This is more commonly written as the dot product of the force and displacement vectors:
$$W = \vec{F} \cdot \vec{d} = Fd \cos\theta$$
Where:

• F is the magnitude of the constant force.
• d is the magnitude of the displacement.
• $\theta$ is the angle between the force vector and the displacement vector.

Key Characteristics

• Scalar Quantity: Work is a scalar quantity, meaning it has magnitude but no direction. It can be positive, negative, or zero.
• Unit: The SI unit of work is the Joule (J). One joule is the work done when a force of one newton displaces an object by one meter in the direction of the force (1 J = 1 N·m).
• Dimensions: The dimensional formula for work is [ML²T⁻²].

Conditions for Work Done

The value and sign of the work done depend on the angle $\theta$:

• Positive Work ($\theta < 90^\circ$): Work is positive when the force (or a component of it) is in the same direction as the displacement. This means the force is helping the motion. Maximum positive work is done when $\theta = 0^\circ$ ($W = Fd$).

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2. Work Done by a Variable Force

Definition

In many real-world scenarios, the force applied to an object is not constant; it changes as the object moves. Examples include the force exerted by a spring, which increases as it is stretched, or the force of a rocket engine, which changes as fuel is consumed.

Calculation Methods

To calculate the work done by a variable force, we cannot simply multiply the force by the distance. Instead, we must account for the changing force.

a) Graphical Method

The work done by a variable force can be found graphically by calculating the area under a Force vs. Displacement graph.

• Procedure: Plot the component of the force parallel to the displacement ($F \cos\theta$) on the y-axis and the displacement (d) on the x-axis.
• Result: The area of the region between the curve and the x-axis gives the total work done.
  
  

b) Interval Method (Calculus)

For a more precise calculation, we can use calculus.

• Procedure:
  1. Divide the total displacement into a large number of infinitesimally small intervals, $\Delta d$.
  2. Over each tiny interval, the force can be considered approximately constant. The small amount of work done, $\Delta W$, in this interval is $\Delta W = F \Delta d \cos\theta$.
  3. The total work done is the sum of the work done in all these small intervals.
• Result (Integral Form): As the intervals become infinitesimally small, this sum becomes an integral. The total work done in moving from an initial position $d_i$ to a final position is:

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Possible Questions/Answer

• Q: What is the difference between the scientific and everyday definitions of "work"?
  A: In everyday language, "work" can refer to any mental or physical effort. In physics, work is only done when a force causes a displacement. Holding a heavy object stationary for a long time may feel like work, but in the physics sense, no work is done on the object because there is no displacement.

• Q: Can work be done by a centripetal force?
  A: No. A centripetal force is always directed perpendicular to the direction of motion (the velocity) of an object in uniform circular motion. Since the angle $\theta$ is 90°, the work done by the centripetal force is always zero ($W = Fd \cos(90^\circ) = 0$).

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Summary

• Work is the energy transferred to or from an object by a force acting over a displacement.
• For a constant force, work is calculated as $W = Fd \cos\theta$.
• For a variable force, work is the area under the Force-Displacement graph or is found by integration.
• Work is a scalar quantity measured in Joules (J) and can be positive, negative, or zero.

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