Our teacher want us to use integrate and use the dot method to solve for this question but I don't know how. Can any turor help me.A particle moves in the xy plane (see the figure below) from the origin to a point having coordinates x = 7m and y = 5m under the influence of a force given by F= 3yi^2 - 5xj.
a) What is the work done on the particle by the force F if it moves along path 1 (ABD)? J
b) What is the work done on the particle by the force F if it moves along path 2 (ACD)? J
c) What is the work done on the particle by the force F if it moves along path 3 (AD)? J
The figure is a square with four side and a daiagonal line from A to D lsimilar to the one below , ignore the zero.
C----------D(7m,5m)
l00000000l
l00000000l
A-----------B
http://cas2.umkc.edu/physics/wrobel/phy240/Homework%20%205.pdf
see prob 7.45 a very similar problem. Analyze it carefully, you will get to the solutions
I spend many hours on analyzing your similar problem but still don't get it. Do you know an easier way bobursley?
To find the work done by the force on the particle along each path, you can use dot product with integration. Here's how you can solve each part:
a) Path 1 (ABD):
To find the work done along this path, you need to calculate the dot product between the force vector and the path vector at each point along the path. The path vector is the tangent vector to the path at each point.
1. Calculate the equation of the path 1 (ABD):
- AB: y = 0, x ranges from 0 to 7
- BD: y ranges from 0 to 5, x = 7
2. Calculate the force vector at each point along the path:
- F = 3y^2i - 5xj
- Plug in the x and y values from the path equation to find the force vector at each point.
3. Calculate the dot product between the force vector and the path vector at each point:
- The dot product is given by F · dr = |F| |dr| cos(theta), where theta is the angle between the force vector and the path vector at each point.
- Since the path is straight, the angle is either 0 or 180 degrees.
- Integrate the dot product over the path to find the total work done:
W = ∫ (F · dr) along path 1
b) Path 2 (ACD):
Follow the same steps as in part a to calculate the work done along path 2. Calculate the path equation, force vector at each point, dot product between the force vector and path vector, and integrate over the path to find the work done:
W = ∫ (F · dr) along path 2
c) Path 3 (AD):
Follow the same steps as in parts a and b to calculate the work done along path 3. Calculate the path equation, force vector at each point, dot product between the force vector and path vector, and integrate over the path to find the work done:
W = ∫ (F · dr) along path 3
Note: The path equations for AB, BD, and AD are straight lines, so you can express them in terms of either x or y. For AC, the path is along the diagonal, so you may need to use a parametric representation of the path.