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Physics / Chapter 2: Motion Along a Straight Line / Section 2-1

Motion — Question 8

Section 2-1 · Physics pages 146–147

Question 8 One moving point cannot reveal every change in a body Three permanently identified marks P, Q, and R lie in that order on a thin strip. The strip stays on one straight laboratory line without turning over. Their locations are recorded against fixed metre marks: The successive position triples (xP , xQ, xR) are (0, 1, 3), (4, 5, 7) and (6, 7.5, 10.5), all in metres. All values are exact model data. A pure translation of a rigid strip moves every material mark by the same amount along the line and preserves all separations. A deformable strip may change separations. No behavior between recorded instants is specified. First P 0 Q 1 R 3 Second P 4 Q 5 R 7 Third P 6 Q 7.5 R 10.5 x Snapshots share one position scale; labels identify the same material marks. (a) Find the P–Q and Q–R separations at each observation. Which successive pair of observed configurations is compatible with pure rigid translation? Which pair rules it out? (b) From the second to the third observation, compare the changes in the three recorded locations. Explain why all three marks moving right does not establish rigid translation. (c) Do the two adjacent separations change by the same factor? Does that prove uniform stretching at every unmarked point of the strip? (d) A simplified tracker records only P. Construct a rigid-strip third configuration that agrees with P’s actual third reading but disagrees with Q and R. State what the one-point description loses. (e) Does the first compatible pair prove that the strip stayed rigid throughout the intervening time?
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Question 8 - Solution One moving point cannot reveal every change in a body First P 0 Q 1 R 3 Second P 4 Q 5 R 7 Third P 6 Q 7.5 R 10.5 x (a) Compare internal separations, not just an overall location. (∆xP , ∆xQ, ∆xR)1→2 = (4, 4, 4) m, (· · · )2→3 = (2, 2.5, 3.5) m. The adjacent separations are (Q −P, R −Q)1,2 = (1, 2) m, (Q −P, R −Q)3 = (1.5, 3) m. The first and second configurations are compatible with rigid translation: every mark shifts by 4.0 m. The second and third are not, because the internal separations change. (b) A common direction is weaker than a common translation. From the second to the third sample, P moves from mark 4.0 to 6.0, Q from 5.0 to 7.5, and R from 7.0 to 10.5. The changes are respectively 2.0, 2.5, 3.5 m to the right. They are all positive but not equal. Motion in a common direction can accompany deformation. (c) A few separation ratios do not specify all local deformation. (Q −P)3 (Q −P)2 = 1.5 1 = (R −Q)3 (R −Q)2 = 3 2. Both adjacent gaps increase by a factor 1.5 . Uniform stretching of the strip is consistent with these samples, but not established at every unmarked point. Different local stretches within either interval could produce the same endpoint separations. Additional material marks or a stronger deformation model are needed. (d) A point description omits internal configuration. Rigid shift matching P: (4, 5, 7) + (2, 2, 2) = (6, 7, 9) m. If the second configuration translated rigidly by 2.0 m, the third would be P = 6.0 m, Q = 7.0 m, R = 9.0 m . P agrees with the actual reading, but Q and R do not. A P-only record cannot distinguish these two configurations; it omits the strip’s changing internal separations. (e) Sampled compatibility does not establish continuous rigidity. Between the first and second observations, the strip could deform and recover its original separations before the second sample. The snapshots admit a rigid-translation history but do not prove that it occurred throughout. A physical rigidity assumption supplies information beyond the sampled locations.

Question 8

One moving point cannot reveal every change in a body

Three permanently identified marks P, Q, and R lie in that order on a thin strip. The strip stays on one straight laboratory line without turning over. Their locations are recorded against fixed metre marks: The successive position triples (xP,xQ,xR)(x_P,x_Q,x_R) are (0,1,3)(0,1,3), (4,5,7)(4,5,7) and (6,7.5,10.5)(6,7.5,10.5), all in metres. All values are exact model data. A pure translation of a rigid strip moves every material mark by the same amount along the line and preserves all separations. A deformable strip may change separations. No behavior between recorded instants is specified.

Diagram shown in the original SVG worksheet.

  1. Find the P–Q and Q–R separations at each observation. Which successive pair of observed configurations is compatible with pure rigid translation? Which pair rules it out?

  2. From the second to the third observation, compare the changes in the three recorded locations. Explain why all three marks moving right does not establish rigid translation.

  3. Do the two adjacent separations change by the same factor? Does that prove uniform stretching at every unmarked point of the strip?

  4. A simplified tracker records only P. Construct a rigid-strip third configuration that agrees with P’s actual third reading but disagrees with Q and R. State what the one-point description loses.

  5. Does the first compatible pair prove that the strip stayed rigid throughout the intervening time?

Question 8 - Solution

One moving point cannot reveal every change in a body

Diagram shown in the original SVG worksheet.

(a) Compare internal separations, not just an overall location.

(ΔxP,ΔxQ,ΔxR)1→2=(4,4,4)m,(⋯)2→3=(2,2.5,3.5)m.(\Delta x_P,\Delta x_Q,\Delta x_R)_{1\to2}=(4,4,4)\ \mathrm{m},\quad(\cdots)_{2\to3}=(2,2.5,3.5)\ \mathrm{m}.

The adjacent separations are (Q−P,R−Q)1,2=(1,2)m,(Q−P,R−Q)3=(1.5,3)m.(Q-P,R-Q)_{1,2}=(1,2)\ \mathrm{m},\quad(Q-P,R-Q)_3=(1.5,3)\ \mathrm{m}. The first and second configurations are compatible with rigid translation: every mark shifts by 4.0 m. The second and third are not, because the internal separations change.

(b) A common direction is weaker than a common translation.

From the second to the third sample, P moves from mark 4.0 to 6.0, Q from 5.0 to 7.5, and R from 7.0 to 10.5. The changes are respectively 2.0,2.5,3.5m\boxed{2.0,\ 2.5,\ 3.5\ \mathrm{m}} to the right. They are all positive but not equal. Motion in a common direction can accompany deformation.

(c) A few separation ratios do not specify all local deformation.

(Q−P)3(Q−P)2=1.51=(R−Q)3(R−Q)2=32.\frac{(Q-P)_3}{(Q-P)_2}=\frac{1.5}{1}=\frac{(R-Q)_3}{(R-Q)_2}=\frac32.

Both adjacent gaps increase by a factor 1.5\boxed{1.5}. Uniform stretching of the strip is consistent with these samples, but not established at every unmarked point. Different local stretches within either interval could produce the same endpoint separations. Additional material marks or a stronger deformation model are needed.

(d) A point description omits internal configuration.

Rigid shift matching P: (4,5,7)+(2,2,2)=(6,7,9)m.\text{Rigid shift matching P: }(4,5,7)+(2,2,2)=(6,7,9)\ \mathrm{m}.

If the second configuration translated rigidly by 2.0 m, the third would be P=6.0m,Q=7.0m,R=9.0m.\boxed{P=6.0\ \mathrm{m},\quad Q=7.0\ \mathrm{m},\quad R=9.0\ \mathrm{m}}. P agrees with the actual reading, but Q and R do not. A P-only record cannot distinguish these two configurations; it omits the strip’s changing internal separations.

(e) Sampled compatibility does not establish continuous rigidity.

Between the first and second observations, the strip could deform and recover its original separations before the second sample. The snapshots admit a rigid-translation history but do not prove that it occurred throughout. A physical rigidity assumption supplies information beyond the sampled locations.

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