A force
F =
(5i − 4j)
N
acts on a particle that undergoes a displacement
Δr = (5i+j)m .
(a) Find the work done by the force on the particle.
(b) What is the angle between F and Δr?
To find the work done by a force on a particle, we can use the formula:
Work = Force * Displacement * cos(angle)
(a) The force F is given as (5i - 4j) N, and the displacement Δr is given as (5i + j) m.
To calculate the work done, we need to find the dot product of the force and displacement vectors. The dot product of two vectors A and B is defined as:
A · B = Ax * Bx + Ay * By
Let's calculate the dot product of the force and displacement vectors:
F · Δr = (5i - 4j) · (5i + j)
= (5 * 5) + (-4 * 1)
= 25 - 4
= 21
Therefore, the work done by the force on the particle is 21 N * m.
(b) The angle between the force F and displacement Δr can be found using the dot product formula:
F · Δr = |F| * |Δr| * cos(angle)
We have already calculated the dot product F · Δr as 21. We know the magnitudes of the force and displacement vectors as:
|F| = sqrt((5)^2 + (-4)^2) = sqrt(25 + 16) = sqrt(41)
|Δr| = sqrt((5)^2 + (1)^2) = sqrt(25 + 1) = sqrt(26)
Now we can rearrange the formula to solve for the angle:
cos(angle) = (F · Δr) / (|F| * |Δr|)
cos(angle) = 21 / (sqrt(41) * sqrt(26))
cos(angle) = 21 / sqrt(1066)
angle = arccos(21 / sqrt(1066))
Using a calculator, we can find the angle to be approximately 19.47 degrees.
Therefore, the angle between F and Δr is approximately 19.47 degrees.