### Physics Work Done By Variable Force

##### Work Done by Variable Force

A variable force is a force whose magnitude changes as it acts on an object. The work done by a variable force is the integral of the force with respect to the displacement of the object. In other words, it is the sum of the work done by the force over each infinitesimal displacement of the object.

##### Mathematical Expression

The mathematical expression for the work done by a variable force is given by:

$$W = \int_a^b F(x) dx$$

where:

- W is the work done by the force (in joules)
- F(x) is the force (in newtons)
- x is the displacement of the object (in meters)
- a and b are the initial and final positions of the object (in meters)

##### Example

Consider a force that varies linearly with displacement, such that:

$$F(x) = kx$$

where k is a constant.

The work done by this force over a displacement of d is given by:

$$W = \int_0^d kx dx = \frac{1}{2}kd^2$$

This shows that the work done by a linearly varying force is proportional to the square of the displacement.

##### Applications

The concept of work done by a variable force has many applications in physics and engineering. Some examples include:

- Calculating the work done by a spring
- Calculating the work done by a gas
- Calculating the work done by a muscle
- Calculating the work done by a machine

Work done by a variable force is an important concept in physics and engineering. It is used to calculate the energy transferred to or from an object by a force that changes in magnitude.

##### Work Done by Variable Force Graph

A variable force is a force whose magnitude or direction changes with time. The work done by a variable force can be calculated using the following formula:

$$W = \int_a^b F(x) dx$$

where:

- $W$ is the work done (in joules)
- $F(x)$ is the force (in newtons) as a function of position $x$ (in meters)
- $a$ and $b$ are the initial and final positions (in meters)

##### Steps to Calculate Work Done by Variable Force Graph

To calculate the work done by a variable force using a graph, follow these steps:

- Divide the $x$-axis into small intervals.
- At each interval, estimate the average force $\overline{F}$ acting on the object.
- Multiply the average force by the change in position $\Delta x$ to get the work done by the force in that interval: $\Delta W = \overline{F} \Delta x$.
- Repeat steps 2 and 3 for each interval.
- Add up the work done in each interval to get the total work done by the force.

##### Example

Consider a force $F(x)$ that varies with position $x$ according to the following graph:

To calculate the work done by this force from $x = 0$ to $x = 5$, we can divide the $x$-axis into five equal intervals of width $\Delta x = 1$. The average force in each interval is:

- Interval 1: $\overline{F}_1 = 2\ N$
- Interval 2: $\overline{F}_2 = 4\ N$
- Interval 3: $\overline{F}_3 = 6\ N$
- Interval 4: $\overline{F}_4 = 8\ N$
- Interval 5: $\overline{F}_5 = 10\ N$

The work done by the force in each interval is:

- Interval 1: $\Delta W_1 = \overline{F}_1 \Delta x = 2\ N \cdot 1\ m = 2\ J$
- Interval 2: $\Delta W_2 = \overline{F}_2 \Delta x = 4\ N \cdot 1\ m = 4\ J$
- Interval 3: $\Delta W_3 = \overline{F}_3 \Delta x = 6\ N \cdot 1\ m = 6\ J$
- Interval 4: $\Delta W_4 = \overline{F}_4 \Delta x = 8\ N \cdot 1\ m = 8\ J$
- Interval 5: $\Delta W_5 = \overline{F}_5 \Delta x = 10\ N \cdot 1\ m = 10\ J$

The total work done by the force is:

$$W = \Delta W_1 + \Delta W_2 + \Delta W_3 + \Delta W_4 + \Delta W_5 = 2\ J + 4\ J + 6\ J + 8\ J + 10\ J = 30\ J$$

Therefore, the work done by the variable force from $x = 0$ to $x = 5$ is 30 joules.

##### Work Done By Variable Force FAQs

##### What is work done by a variable force?

Work done by a variable force is the amount of energy transferred to or from an object by a force that changes in magnitude or direction. It is calculated as the integral of the force with respect to the displacement of the object.

##### How do you calculate the work done by a variable force?

The work done by a variable force can be calculated using the following formula:

$$ W = ∫ F(x) dx $$

where:

- W is the work done (in joules)
- F(x) is the force (in newtons)
- x is the displacement (in meters)

##### What are some examples of work done by a variable force?

Some examples of work done by a variable force include:

- The work done by a person pushing a lawnmower
- The work done by a car engine accelerating a car
- The work done by a spring when it is stretched or compressed

##### What is the difference between work done by a constant force and work done by a variable force?

The work done by a constant force is equal to the product of the force and the displacement of the object. The work done by a variable force, on the other hand, is equal to the integral of the force with respect to the displacement of the object.

##### What are some of the applications of work done by a variable force?

Work done by a variable force has a number of applications, including:

- Calculating the energy efficiency of machines
- Designing engines and other mechanical devices
- Analyzing the motion of objects in space

##### Conclusion

Work done by a variable force is an important concept in physics. It is used to calculate the amount of energy transferred to or from an object by a force that changes in magnitude or direction. Work done by a variable force has a number of applications, including calculating the energy efficiency of machines, designing engines and other mechanical devices, and analyzing the motion of objects in space.