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Work — Question 9

Section 7-3 · Physics pages 908–909

Question 9 Third-law forces need not have cancelling works Two particles A and B interact through constant forces: B exerts +10 N on A and A exerts −10 N on B along x. During the selected interval A moves +2 m and B moves +5 m. Other external agents may be present. A B ∆xA = +2 m ∆xB = +5 m FA = +10 N FB = −10 N Equal and opposite forces act through different point displacements. (a) Find work by the interaction on each particle. (b) Find the sum of these two works. (c) What relation between the two displacements would make their interaction works cancel?
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Question 9 - Solution Third-law forces need not have cancelling works A B ∆xA = +2 m ∆xB = +5 m FA = +10 N FB = −10 N (a) Use the displacement of the correct body. The governing relation is WA + WB =⃗FA · (∆⃗rA −∆⃗rB) WA = 10(2) = 20 J, WB = −10(5) = −50 J. (b) Add the scalar results. WA + WB = −30 J. Equal and opposite forces act on different particles whose displacements need not match. (c) Derive the cancellation condition. For these collinear forces, WA + WB = 10(∆xA −∆xB). It vanishes if the displacements are equal. More generally, only their difference projected along the interaction force must vanish.

Question 9

Third-law forces need not have cancelling works

Two particles A and B interact through constant forces: B exerts +10 N on A and A exerts −10-10 N on B along x. During the selected interval A moves +2 m and B moves +5 m. Other external agents may be present.

Diagram shown in the original SVG worksheet.

  1. Find work by the interaction on each particle.

  2. Find the sum of these two works.

  3. What relation between the two displacements would make their interaction works cancel?

Question 9 - Solution

Third-law forces need not have cancelling works

Diagram shown in the original SVG worksheet.

(a) Use the displacement of the correct body.

The governing relation is WA+WB=F→A⋅(Δr→A−Δr→B)\boxed{W_A+W_B=\vec F_A\cdot(\Delta\vec r_A-\Delta\vec r_B)}

WA=10(2)=20J,WB=−10(5)=−50J.W_A=10(2)=20\ \mathrm J,\qquad W_B=-10(5)=-50\ \mathrm J.

(b) Add the scalar results.

WA+WB=−30J.W_A+W_B=-30\ \mathrm J. Equal and opposite forces act on different particles whose displacements need not match.

(c) Derive the cancellation condition.

For these collinear forces, WA+WB=10(ΔxA−ΔxB).W_A+W_B=10(\Delta x_A-\Delta x_B). It vanishes if the displacements are equal. More generally, only their difference projected along the interaction force must vanish.

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