A force acts in the direction $(4,2)$, moving an object along a straight path from $(2,-4)$ to $(10,1)$. How much work is done by the force?
A. 12 joules
B. 42 joules
C. 48 joules
D. 78 joules
Answer (2)
The work done by the force acting on the object as it moves from (2, -4) to (10, 1) is calculated using the dot product of the force vector (4, 2) and the displacement vector (8, 5). The total work done is 42 joules. Thus, the correct answer is option B: 42 joules.
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Calculate the displacement vector by subtracting the initial position from the final position: ( 10 , 1 ) − ( 2 , − 4 ) = ( 8 , 5 ) .
Calculate the dot product of the force vector ( 4 , 2 ) and the displacement vector ( 8 , 5 ) .
The work done is equal to the dot product: ( 4 × 8 ) + ( 2 × 5 ) = 32 + 10 = 42 .
The work done by the force is 42 joules.
Explanation
Problem Analysis The problem asks us to find the work done by a force acting on an object as it moves along a straight path. We are given the force vector and the initial and final positions of the object.
Calculate Displacement Vector First, we need to find the displacement vector. The displacement vector is the difference between the final and initial positions. So, we subtract the initial position ( 2 , − 4 ) from the final position ( 10 , 1 ) :
( 10 , 1 ) − ( 2 , − 4 ) = ( 10 − 2 , 1 − ( − 4 )) = ( 8 , 5 ) Thus, the displacement vector is ( 8 , 5 ) .
Calculate Work Done Next, we need to calculate the work done by the force. The work done is the dot product of the force vector and the displacement vector. The force vector is ( 4 , 2 ) and the displacement vector is ( 8 , 5 ) . The dot product is calculated as follows: ( 4 , 2 ) ⋅ ( 8 , 5 ) = ( 4 × 8 ) + ( 2 × 5 ) = 32 + 10 = 42 Therefore, the work done is 42 joules.
Final Answer The work done by the force is 42 joules.
Examples
Imagine you're pushing a box across a floor. The force you apply to the box and the distance the box moves determine the amount of work you've done. This concept is crucial in physics and engineering for calculating energy transfer in various systems, from simple machines to complex mechanical systems.
