Review Derivation: Physics Derivation Graph

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pdg_app/to_review_derivation sympy_validate_step.validate_step: AssertionError 0004482244

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Review equation of motion for a spring

abstract:
reference:

step	inference rule	input	feed	output	step validity (as per SymPy)
1	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			5345738321 $F = m a$	no validation is available for declarations
2	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			4428528271 $F_{\rm spring} = -k x$	no validation is available for declarations
3	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			6831694380 $a = \frac{d^2 x}{dt^2}$	no validation is available for declarations
4	0000111355: LHS of expr 1 equals LHS of expr 2 number of inputs: 2; feeds: 0; outputs: 1 LHS of Eq.~\ref{eq:#1} is equal to LHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	4428528271 $F_{\rm spring} = -k x$ 5345738321 $F = m a$		2334518266 $m a = -k x$	input diff is pdg0004183 - pdg0004202 diff is -pdg0001356pdg0004037 - pdg0005156pdg0009140 diff is pdg0001356pdg0004037 + pdg0005156pdg0009140
5	0000111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	2334518266 $m a = -k x$	3634715785 $m$	8655294002 $a = -\frac{k}{m}x$	valid
6	0000111355: LHS of expr 1 equals LHS of expr 2 number of inputs: 2; feeds: 0; outputs: 1 LHS of Eq.~\ref{eq:#1} is equal to LHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	6831694380 $a = \frac{d^2 x}{dt^2}$ 8655294002 $a = -\frac{k}{m}x$		8991236357 $\frac{d^2 x}{dt^2} = -\frac{k}{m} x$	input diff is a - pdg0009140 diff is d*2(-pdg0004037 + x)/dt**2 diff is 0
Note about step 7: what, when differentiated twice, yields a negative of itself? cosine
7	0000111237: declare guess solution number of inputs: 1; feeds: 0; outputs: 1 Judicious choice as a guessed solution to Eq.~\ref{eq:#1} is Eq.~\ref{eq:#2},	8991236357 $\frac{d^2 x}{dt^2} = -\frac{k}{m} x$		5415824175 $x(t) = A \cos(\omega t)$	no validation is available for declarations
8	0000111649: differentiate with respect to number of inputs: 1; feeds: 1; outputs: 1 Differentiate Eq.~\ref{eq:#2} with respect to $#1$; yields Eq.~\ref{eq:#3}.	5415824175 $x(t) = A \cos(\omega t)$	5846177002 $t$	7652131521 $\frac{dx}{dt} = -A \omega \sin (\omega t)$	LHS diff is 0 RHS diff is pdg0002321pdg0009885sin(pdg0001467*pdg0002321)
9	0000111649: differentiate with respect to number of inputs: 1; feeds: 1; outputs: 1 Differentiate Eq.~\ref{eq:#2} with respect to $#1$; yields Eq.~\ref{eq:#3}.	7652131521 $\frac{dx}{dt} = -A \omega \sin (\omega t)$	1451839362 $t$	5945893986 $\frac{d^2 x}{dt^2} = -A \omega^2 \cos(\omega t)$	LHS diff is -d*2x/dt**2 RHS diff is 0
10	0000111355: LHS of expr 1 equals LHS of expr 2 number of inputs: 2; feeds: 0; outputs: 1 LHS of Eq.~\ref{eq:#1} is equal to LHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	8991236357 $\frac{d^2 x}{dt^2} = -\frac{k}{m} x$ 5945893986 $\frac{d^2 x}{dt^2} = -A \omega^2 \cos(\omega t)$		1772973171 $-\frac{k}{m} x = -A \omega^2 \cos(\omega t)$	input diff is d*2(pdg0004037 - x)/dt*2 diff is (kx - pdg0001356pdg0004037)/pdg0005156 diff is pdg00023212(Acos(pdg0002321pdg0009491) - pdg0009885cos(pdg0001467pdg0002321))
11	0000111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	1772973171 $-\frac{k}{m} x = -A \omega^2 \cos(\omega t)$ 5415824175 $x(t) = A \cos(\omega t)$		2148049269 $-\frac{k}{m} A \cos(\omega t) = -A \omega^2 \cos(\omega t)$	LHS diff is Akcos(pdg0002321pdg0009491)/pdg0005156 + x(pdg0001467) RHS diff is Apdg0002321*2cos(pdg0002321pdg0009491) + pdg0009885cos(pdg0001467*pdg0002321)
12	0000111182: multiply both sides by number of inputs: 1; feeds: 1; outputs: 1 Multiply both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	2148049269 $-\frac{k}{m} A \cos(\omega t) = -A \omega^2 \cos(\omega t)$	7473576008 $\frac{-1}{A \cos(\omega t)}$	1931103031 $\frac{k}{m} = \omega^2$	LHS arithmetic error. Diff: -(Akpdg0001467cos(pdg0002321pdg0009491) + pdg0001356)/pdg0005156
13	0000111524: square root both sides number of inputs: 1; feeds: 0; outputs: 2 Take the square root of both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2} and Eq.~\ref{eq:#3}.	1931103031 $\frac{k}{m} = \omega^2$		1888494137 $-\sqrt{\frac{k}{m}} = \omega$ 1784114349 $\sqrt{\frac{k}{m}} = \omega$
14	0000111634: substitute RHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute RHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	5415824175 $x(t) = A \cos(\omega t)$ 1784114349 $\sqrt{\frac{k}{m}} = \omega$		6908055431 $x(t) = A \cos\left(\frac{k}{m} t\right)$	LHS diff is sqrt(pdg0001356/pdg0005156) - x(pdg0001467) RHS diff is pdg0002321 - pdg0009885cos(kpdg0001467/pdg0005156)
15	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	6908055431 $x(t) = A \cos\left(\frac{k}{m} t\right)$			no validation is available for declarations

Symbols used in equation of motion for a spring

0000002321: $ \omega $
0000009885: $ A $
0000009140: $ a $
0000004202: $ F $
0000004183: $ F_{\rm spring} $
0000001356: $ k $
0000005156: $ m $
0000001467: $ t $
0000009491: $ T $
0000004037: $ x $

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0.018 seconds for pdg_app/to_review_derivation 2a614075-087e-4e20-bf5a-1b529245e187
0.02 seconds for compute/get_dict_of_steps_in_derivation: steps_in_this_derivationd3979753-0f3a-4919-9c4f-4bb1261fb33f
0.007 seconds for compute/input_feed_output_infrule_for_step: get_inference_rule_connected_to_step_IDd3979753-0f3a-4919-9c4f-4bb1261fb33f
0.03 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type HAS_INPUTd3979753-0f3a-4919-9c4f-4bb1261fb33f
0.006 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_FEEDd3979753-0f3a-4919-9c4f-4bb1261fb33f
0.013 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_OUTPUTd3979753-0f3a-4919-9c4f-4bb1261fb33f
0.009 seconds for compute/get_dict_of_steps_in_derivation: get_sequence_index_for_stepd3979753-0f3a-4919-9c4f-4bb1261fb33f
0.019 seconds for pdg_app/ 2a614075-087e-4e20-bf5a-1b529245e187

step	inference rule	input	feed	output	step validity (as per SymPy)
1	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			5345738321 \(F = m a\)	no validation is available for declarations
2	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			4428528271 \(F_{\rm spring} = -k x\)	no validation is available for declarations
3	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			6831694380 \(a = \frac{d^2 x}{dt^2}\)	no validation is available for declarations
4	0000111355: LHS of expr 1 equals LHS of expr 2 number of inputs: 2; feeds: 0; outputs: 1 LHS of Eq.~\ref{eq:#1} is equal to LHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	4428528271 \(F_{\rm spring} = -k x\) 5345738321 \(F = m a\)		2334518266 \(m a = -k x\)	input diff is pdg0004183 - pdg0004202 diff is -pdg0001356pdg0004037 - pdg0005156pdg0009140 diff is pdg0001356pdg0004037 + pdg0005156pdg0009140
5	0000111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	2334518266 \(m a = -k x\)	3634715785 \(m\)	8655294002 \(a = -\frac{k}{m}x\)	valid
6	0000111355: LHS of expr 1 equals LHS of expr 2 number of inputs: 2; feeds: 0; outputs: 1 LHS of Eq.~\ref{eq:#1} is equal to LHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	6831694380 \(a = \frac{d^2 x}{dt^2}\) 8655294002 \(a = -\frac{k}{m}x\)		8991236357 \(\frac{d^2 x}{dt^2} = -\frac{k}{m} x\)	input diff is a - pdg0009140 diff is d*2(-pdg0004037 + x)/dt**2 diff is 0
Note about step 7: what, when differentiated twice, yields a negative of itself? cosine
7	0000111237: declare guess solution number of inputs: 1; feeds: 0; outputs: 1 Judicious choice as a guessed solution to Eq.~\ref{eq:#1} is Eq.~\ref{eq:#2},	8991236357 \(\frac{d^2 x}{dt^2} = -\frac{k}{m} x\)		5415824175 \(x(t) = A \cos(\omega t)\)	no validation is available for declarations
8	0000111649: differentiate with respect to number of inputs: 1; feeds: 1; outputs: 1 Differentiate Eq.~\ref{eq:#2} with respect to $#1$; yields Eq.~\ref{eq:#3}.	5415824175 \(x(t) = A \cos(\omega t)\)	5846177002 \(t\)	7652131521 \(\frac{dx}{dt} = -A \omega \sin (\omega t)\)	LHS diff is 0 RHS diff is pdg0002321pdg0009885sin(pdg0001467*pdg0002321)
9	0000111649: differentiate with respect to number of inputs: 1; feeds: 1; outputs: 1 Differentiate Eq.~\ref{eq:#2} with respect to $#1$; yields Eq.~\ref{eq:#3}.	7652131521 \(\frac{dx}{dt} = -A \omega \sin (\omega t)\)	1451839362 \(t\)	5945893986 \(\frac{d^2 x}{dt^2} = -A \omega^2 \cos(\omega t)\)	LHS diff is -d*2x/dt**2 RHS diff is 0
10	0000111355: LHS of expr 1 equals LHS of expr 2 number of inputs: 2; feeds: 0; outputs: 1 LHS of Eq.~\ref{eq:#1} is equal to LHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	8991236357 \(\frac{d^2 x}{dt^2} = -\frac{k}{m} x\) 5945893986 \(\frac{d^2 x}{dt^2} = -A \omega^2 \cos(\omega t)\)		1772973171 \(-\frac{k}{m} x = -A \omega^2 \cos(\omega t)\)	input diff is d*2(pdg0004037 - x)/dt*2 diff is (kx - pdg0001356pdg0004037)/pdg0005156 diff is pdg00023212(Acos(pdg0002321pdg0009491) - pdg0009885cos(pdg0001467pdg0002321))
11	0000111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	1772973171 \(-\frac{k}{m} x = -A \omega^2 \cos(\omega t)\) 5415824175 \(x(t) = A \cos(\omega t)\)		2148049269 \(-\frac{k}{m} A \cos(\omega t) = -A \omega^2 \cos(\omega t)\)	LHS diff is Akcos(pdg0002321pdg0009491)/pdg0005156 + x(pdg0001467) RHS diff is Apdg0002321*2cos(pdg0002321pdg0009491) + pdg0009885cos(pdg0001467*pdg0002321)
12	0000111182: multiply both sides by number of inputs: 1; feeds: 1; outputs: 1 Multiply both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	2148049269 \(-\frac{k}{m} A \cos(\omega t) = -A \omega^2 \cos(\omega t)\)	7473576008 \(\frac{-1}{A \cos(\omega t)}\)	1931103031 \(\frac{k}{m} = \omega^2\)	LHS arithmetic error. Diff: -(Akpdg0001467cos(pdg0002321pdg0009491) + pdg0001356)/pdg0005156
13	0000111524: square root both sides number of inputs: 1; feeds: 0; outputs: 2 Take the square root of both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2} and Eq.~\ref{eq:#3}.	1931103031 \(\frac{k}{m} = \omega^2\)		1888494137 \(-\sqrt{\frac{k}{m}} = \omega\) 1784114349 \(\sqrt{\frac{k}{m}} = \omega\)
14	0000111634: substitute RHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute RHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	5415824175 \(x(t) = A \cos(\omega t)\) 1784114349 \(\sqrt{\frac{k}{m}} = \omega\)		6908055431 \(x(t) = A \cos\left(\frac{k}{m} t\right)\)	LHS diff is sqrt(pdg0001356/pdg0005156) - x(pdg0001467) RHS diff is pdg0002321 - pdg0009885cos(kpdg0001467/pdg0005156)
15	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	6908055431 \(x(t) = A \cos\left(\frac{k}{m} t\right)\)			no validation is available for declarations