# Work done by a force with a given equation

• trogdor5
In summary, the conversation is about finding the work done by a given force in moving a particle from one position to another along a specific path. The work is calculated by taking the integral of the force with respect to distance, where both are expressed as vectors. The vectors are represented by i and j coordinates, with x and y being the magnitudes in those directions. To solve the problem, the first step is to express the position vector in terms of x and y, and then take the dot product with the vector form of the force.
trogdor5

## Homework Statement

A force acts on a particle and is given by the following expression:
F(x,y)=2x^3 y^2 i+3xy^3 j
What is the work done by this force in moving the particle from a position (x,y) = (0,0) to (4,2) along the path given by the curve y=√x ?

## Homework Equations

I know the work is the integral of the Force

## The Attempt at a Solution

Honestly, very confused. I don't know how to deal with the i and j values and I have no clue how to handle the work done along the path of a different curve. Any help is appreciated!

Yes, work is the integral of the force wrt distance, each being a vector. So the integrand and differential element look like F.dp, the dot being the scalar product of the two vectors. (I've used p for position vector to avoid confusion with the x scalar in the question.) So you need to express the vector distance element along the path in i and j coordinates. When the particle moves a distance dy in the y direction, how far does it go in the x direction?

I kind of understand what you're saying, but not really. I don't even think I understand what the question is asking to be honest. I don't understand how there can be x and y but also i and j coordinate systems.

The i and j are unit vectors representing the x and y directions respectively. X and y themselves are magnitudes of position in those directions. I.e. the position vector of the particle at time t is x(t)i + y(t)j.

I'm honestly trying to do it but since my math isn't that strong and I've never seen a problem like this I'm just having problems. Can you do the first step for me or walk me through a bit more step-by-step? I'm just having extreme difficulty

y2 = x; 2ydy = dx
A small step in position = idx + jdy = 2iydy + jdy
You know the vector form of F. Take the dot product of this with position differential above.

## 1. What is work done by a force?

Work done by a force is a physical quantity that measures the energy transfer that occurs when a force is applied to an object and causes it to move a certain distance.

## 2. How is work calculated?

Work is calculated by multiplying the magnitude of the force applied to the object by the distance the object moves in the direction of the force. This can be represented mathematically as W = F * d.

## 3. What is the unit of work?

The unit of work is joule (J) in the International System of Units (SI). It can also be expressed in other units such as foot-pound (ft-lb) or calorie (cal).

## 4. Can work be negative?

Yes, work can be negative if the force and displacement are in opposite directions. This indicates that the force is doing work against the motion of the object, such as in the case of friction or a force applied in the opposite direction of motion.

## 5. How does the equation for work relate to the graph of a force-distance curve?

The work done by a force is equal to the area under the force-distance curve. This means that the total work done is represented by the shaded region on the graph, where the x-axis represents distance and the y-axis represents force.

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