A particle is displaced slowly from (2, 0, 3) m to (1, 1, 2) m. If one of the forces acting on the particle is (i - j + 3k) N, total work done (in J) by the remaining forces acting... A particle is displaced slowly from (2, 0, 3) m to (1, 1, 2) m. If one of the forces acting on the particle is (i - j + 3k) N, total work done (in J) by the remaining forces acting on the particle is?
Understand the Problem
The question is asking to calculate the total work done on a particle that is displaced from one point to another while a force is applied. It involves understanding the concepts of work, force, and displacement in the context of physics.
Answer
The total work done by the remaining forces acting on the particle is \( 5 \, \text{J} \).
Answer for screen readers
The total work done by the remaining forces acting on the particle is ( 5 , \text{J} ).
Steps to Solve
- Calculate displacement vector
To calculate the displacement vector, subtract the initial position vector from the final position vector.
Initial position: ( \mathbf{r_1} = (2, 0, 3) ) m
Final position: ( \mathbf{r_2} = (1, 1, 2) ) m
Displacement vector ( \mathbf{d} ) can be calculated as:
$$ \mathbf{d} = \mathbf{r_2} - \mathbf{r_1} = (1 - 2, 1 - 0, 2 - 3) = (-1, 1, -1) , \text{m} $$
- Calculate the work done by the applied force
Work done ( W ) by the force ( \mathbf{F} ) is given by the dot product of the force and the displacement vector:
Given force: ( \mathbf{F} = (1, -1, 3) ) N
So,
$$ W = \mathbf{F} \cdot \mathbf{d} $$
Calculating the dot product:
$$ W = (1)(-1) + (-1)(1) + (3)(-1) = -1 - 1 - 3 = -5 , \text{J} $$
- Determine the total work done by remaining forces
Since the particle is displaced slowly, the total work done by the net force acting on the particle must be zero (the forces that act against each other), thus:
$$ W_{\text{remaining}} = -W_{\text{applied}} $$
$$ W_{\text{remaining}} = -(-5) = 5 , \text{J} $$
The total work done by the remaining forces acting on the particle is ( 5 , \text{J} ).
More Information
In this problem, because the particle is moved slowly, it implies that the net work done on it is zero. Therefore, the work done by external forces must balance the work done by the force given, leading to the calculated total work for the remaining forces.
Tips
- Failing to properly compute the displacement vector by not keeping track of initial and final positions.
- Confusing the direction of forces and displacement when calculating the dot product.
- Not recognizing that a particle moving slowly suggests net work done on it is zero, leading to a misunderstanding of remaining forces.
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