Find the work done by a 15 N force in the direction of the vector (1,2,2) when it moves a particle from O(0,0,0) to P(1,-3,4) and then from P to A(7,2,5)
Can somebody explain to me how to do this question i don't understand it.
To find the work done by the force in the given scenario, we need to use the formula for work:
Work = Force * Displacement * cos(theta)
Here's how you can do it step by step:
1. Determine the displacement vector from O to P:
To find the displacement vector from O(0,0,0) to P(1,-3,4), subtract the coordinates of O from the coordinates of P:
Displacement vector from O to P = (1-0, -3-0, 4-0) = (1, -3, 4)
2. Calculate the magnitude of the displacement vector from O to P:
The magnitude of a vector AB = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2).
In this case, we have:
Magnitude of displacement from O to P = sqrt((1-0)^2 + (-3-0)^2 + (4-0)^2)
= sqrt(1 + 9 + 16)
= sqrt(26)
3. Calculate the unit vector in the direction of the displacement from O to P:
The unit vector in the direction of a vector AB = (AB / |AB|), where |AB| is the magnitude of vector AB.
In this case, the unit vector in the direction of the displacement from O to P = (1, -3, 4) / sqrt(26)
4. Calculate the dot product between the force vector and the unit vector in the direction of the displacement from O to P:
The dot product of two vectors A and B = |A| * |B| * cos(theta), where theta is the angle between the two vectors.
In our case, the force vector is given as 15 N, and the unit vector in the direction of the displacement from O to P is calculated in step 3 above.
So, the dot product = 15 * sqrt(26) * cos(theta)
5. Calculate theta (angle between the force vector and the unit vector):
To find theta, we can use the dot product formula mentioned in step 4. Rearrange the formula as:
cos(theta) = dot product / (|A| * |B|)
In our case, since we have already calculated the dot product in step 4, we can rearrange the formula to find theta:
theta = arccos(dot product / (15 * sqrt(26)))
6. Finally, calculate the work done:
The work done by the force in the direction of the displacement from O to P can be found using the formula:
Work = Force * Displacement * cos(theta)
= 15 * sqrt(26) * cos(theta)
Repeat the above steps for the displacement from P to A to find the work done during that process.
Note: The cosine of theta becomes negative if the angle between the force and displacement vectors is more than 90 degrees. In that case, the work done will be negative, indicating that the force is acting against the motion.