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review derivation: equation of motion for a spring

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
1 declare initial expr
  1. 5345738321; locally 3065061:
    \(F = m a\)
    \(pdg_{4202} = pdg_{5156} pdg_{9140}\)
no validation is available for declarations 5345738321:
5345738321:
2 declare initial expr
  1. 4428528271; locally 2664105:
    \(F_{\rm{spring}} = -k x\)
    \(pdg_{4183} = - pdg_{1356} pdg_{4037}\)
no validation is available for declarations 4428528271:
4428528271:
3 declare initial expr
  1. 6831694380; locally 9761157:
    \(a = \frac{d^2 x}{dt^2}\)
    \(a = \frac{d^{2} x}{dt^{2}}\)
no validation is available for declarations 6831694380:
6831694380:
4 LHS of expr 1 equals LHS of expr 2
  1. 5345738321; locally 3065061:
    \(F = m a\)
    \(pdg_{4202} = pdg_{5156} pdg_{9140}\)
  2. 4428528271; locally 2664105:
    \(F_{\rm{spring}} = -k x\)
    \(pdg_{4183} = - pdg_{1356} pdg_{4037}\)
  1. 2334518266; locally 7273319:
    \(m a = -k x\)
    \(pdg_{5156} pdg_{9140} = - pdg_{1356} pdg_{4037}\)
input diff is -pdg4183 + pdg4202 diff is 0 diff is 0 5345738321:
4428528271:
2334518266:
5345738321:
4428528271:
2334518266:
5 divide both sides by
  1. 2334518266; locally 7273319:
    \(m a = -k x\)
    \(pdg_{5156} pdg_{9140} = - pdg_{1356} pdg_{4037}\)
  1. 3634715785:
    \(m\)
    \(pdg_{5156}\)
  1. 8655294002; locally 5403312:
    \(a = -\frac{k}{m}x\)
    \(pdg_{9140} = - \frac{pdg_{1356} pdg_{4037}}{pdg_{5156}}\)
valid 2334518266:
8655294002:
2334518266:
8655294002:
6 LHS of expr 1 equals LHS of expr 2
  1. 8655294002; locally 5403312:
    \(a = -\frac{k}{m}x\)
    \(pdg_{9140} = - \frac{pdg_{1356} pdg_{4037}}{pdg_{5156}}\)
  2. 6831694380; locally 9761157:
    \(a = \frac{d^2 x}{dt^2}\)
    \(a = \frac{d^{2} x}{dt^{2}}\)
  1. 8991236357; locally 4154219:
    \(\frac{d^2 x}{dt^2} = -\frac{k}{m} x\)
    \(\frac{d^{2} pdg_{4037}}{dt^{2}} = - \frac{pdg_{1356} pdg_{4037}}{pdg_{5156}}\)
input diff is -a + pdg9140 diff is d**2*pdg4037/dt**2 + pdg1356*pdg4037/pdg5156 diff is -d**2*x/dt**2 - pdg1356*pdg4037/pdg5156 8655294002:
6831694380:
8991236357:
8655294002:
6831694380:
8991236357:
7 declare guess solution
  1. 8991236357; locally 4154219:
    \(\frac{d^2 x}{dt^2} = -\frac{k}{m} x\)
    \(\frac{d^{2} pdg_{4037}}{dt^{2}} = - \frac{pdg_{1356} pdg_{4037}}{pdg_{5156}}\)
  1. 5415824175; locally 2877569:
    \(x(t) = A \cos(\omega t)\)
    \(x{\left(pdg_{1467} \right)} = pdg_{9885} \cos{\left(pdg_{1467} pdg_{2321} \right)}\)
no validation is available for declarations 8991236357:
5415824175:
8991236357:
5415824175:
what, when differentiated twice, yields a negative of itself? cosine
8 differentiate with respect to
  1. 5415824175; locally 2877569:
    \(x(t) = A \cos(\omega t)\)
    \(x{\left(pdg_{1467} \right)} = pdg_{9885} \cos{\left(pdg_{1467} pdg_{2321} \right)}\)
  1. 5846177002:
    \(t\)
    \(t\)
  1. 7652131521; locally 4463004:
    \(\frac{dx}{dt} = -A \omega \sin (\omega t)\)
    \(\frac{d}{d pdg_{1467}} pdg_{4037} = - pdg_{2321} pdg_{9885} \sin{\left(pdg_{1467} pdg_{2321} \right)}\)
no check performed 5415824175:
7652131521:
5415824175:
7652131521:
9 differentiate with respect to
  1. 7652131521; locally 4463004:
    \(\frac{dx}{dt} = -A \omega \sin (\omega t)\)
    \(\frac{d}{d pdg_{1467}} pdg_{4037} = - pdg_{2321} pdg_{9885} \sin{\left(pdg_{1467} pdg_{2321} \right)}\)
  1. 1451839362:
    \(t\)
    \(pdg_{1467}\)
  1. 5945893986; locally 4836115:
    \(\frac{d^2 x}{dt^2} = -A \omega^2 \cos(\omega t)\)
    \(\frac{d^{2} x}{dt^{2}} = - pdg_{2321}^{2} pdg_{9885} \cos{\left(pdg_{1467} pdg_{2321} \right)}\)
no check performed 7652131521:
5945893986:
7652131521:
5945893986:
10 LHS of expr 1 equals LHS of expr 2
  1. 5945893986; locally 4836115:
    \(\frac{d^2 x}{dt^2} = -A \omega^2 \cos(\omega t)\)
    \(\frac{d^{2} x}{dt^{2}} = - pdg_{2321}^{2} pdg_{9885} \cos{\left(pdg_{1467} pdg_{2321} \right)}\)
  2. 8991236357; locally 4154219:
    \(\frac{d^2 x}{dt^2} = -\frac{k}{m} x\)
    \(\frac{d^{2} pdg_{4037}}{dt^{2}} = - \frac{pdg_{1356} pdg_{4037}}{pdg_{5156}}\)
  1. 1772973171; locally 7792692:
    \(-\frac{k}{m} x = -A \omega^2 \cos(\omega t)\)
    \(- \frac{k x}{m} = - A \omega^{2} \cos{\left(\omega t \right)}\)
input diff is d**2*(-pdg4037 + x)/dt**2 diff is -k*x/m + pdg2321**2*pdg9885*cos(pdg1467*pdg2321) diff is -A*omega**2*cos(omega*t) + pdg1356*pdg4037/pdg5156 5945893986:
8991236357:
1772973171:
5945893986:
8991236357:
1772973171:
11 substitute LHS of expr 1 into expr 2
  1. 5415824175; locally 2877569:
    \(x(t) = A \cos(\omega t)\)
    \(x{\left(pdg_{1467} \right)} = pdg_{9885} \cos{\left(pdg_{1467} pdg_{2321} \right)}\)
  2. 1772973171; locally 7792692:
    \(-\frac{k}{m} x = -A \omega^2 \cos(\omega t)\)
    \(- \frac{k x}{m} = - A \omega^{2} \cos{\left(\omega t \right)}\)
  1. 2148049269; locally 7745098:
    \(-\frac{k}{m} A \cos(\omega t) = -A \omega^2 \cos(\omega t)\)
    \(- \frac{A k \cos{\left(\omega t \right)}}{m} = - A \omega^{2} \cos{\left(\omega t \right)}\)
LHS diff is k*(A*cos(omega*t) - x)/m RHS diff is 0 5415824175:
1772973171:
2148049269:
5415824175:
1772973171:
2148049269:
12 multiply both sides by
  1. 2148049269; locally 7745098:
    \(-\frac{k}{m} A \cos(\omega t) = -A \omega^2 \cos(\omega t)\)
    \(- \frac{A k \cos{\left(\omega t \right)}}{m} = - A \omega^{2} \cos{\left(\omega t \right)}\)
  1. 7473576008:
    \(\frac{-1}{A \cos(\omega t)}\)
    \(a e^{2} l o r^{3} t x\)
  1. 1931103031; locally 8360924:
    \(\frac{k}{m} = \omega^2\)
    \(\frac{pdg_{1356}}{pdg_{5156}} = pdg_{2321}^{2}\)
LHS diff is -A*a*e**2*k*l*o*r**3*t*x*cos(omega*t)/m - pdg1356/pdg5156 RHS diff is -A*a*e**2*l*o*omega**2*r**3*t*x*cos(omega*t) - pdg2321**2 2148049269:
1931103031:
2148049269:
1931103031:
13 square root both sides
  1. 1931103031; locally 8360924:
    \(\frac{k}{m} = \omega^2\)
    \(\frac{pdg_{1356}}{pdg_{5156}} = pdg_{2321}^{2}\)
  1. 1784114349; locally 7243628:
    \(\sqrt{\frac{k}{m}} = \omega\)
    \(\sqrt{\frac{pdg_{1356}}{pdg_{5156}}} = pdg_{2321}\)
  2. 1888494137; locally 2051755:
    \(-\sqrt{\frac{k}{m}} = \omega\)
    \(- \sqrt{\frac{pdg_{1356}}{pdg_{5156}}} = pdg_{2321}\)
no check performed 1931103031:
1784114349:
1888494137:
1931103031:
1784114349:
1888494137:
14 substitute RHS of expr 1 into expr 2
  1. 1784114349; locally 7243628:
    \(\sqrt{\frac{k}{m}} = \omega\)
    \(\sqrt{\frac{pdg_{1356}}{pdg_{5156}}} = pdg_{2321}\)
  2. 5415824175; locally 2877569:
    \(x(t) = A \cos(\omega t)\)
    \(x{\left(pdg_{1467} \right)} = pdg_{9885} \cos{\left(pdg_{1467} pdg_{2321} \right)}\)
  1. 6908055431; locally 5872898:
    \(x(t) = A \cos\left(\frac{k}{m} t\right)\)
    \(x{\left(pdg_{1467} \right)} = pdg_{9885} \cos{\left(\frac{k pdg_{1467}}{pdg_{5156}} \right)}\)
LHS diff is 0 RHS diff is pdg9885*(cos(pdg1467*sqrt(pdg1356/pdg5156)) - cos(k*pdg1467/pdg5156)) 1784114349:
5415824175:
6908055431: error for dim with 6908055431
1784114349:
5415824175:
6908055431: N/A
15 declare final expr
  1. 6908055431; locally 5872898:
    \(x(t) = A \cos\left(\frac{k}{m} t\right)\)
    \(x{\left(pdg_{1467} \right)} = pdg_{9885} \cos{\left(\frac{k pdg_{1467}}{pdg_{5156}} \right)}\)
no validation is available for declarations 6908055431: error for dim with 6908055431
6908055431: N/A
Physics Derivation Graph: Steps for equation of motion for a spring

Symbols for this derivation

See also all 215 symbols
symbol ID category latex scope dimension name value Used in derivations references
4037 variable x
\(x\)
['real']
  • length: 1
position 52
9885 variable A
\(A\)
real dimensionless amplitude 4
1467 variable t
\(t\)
['real']
  • time: 1
time 120
4202 variable F
\(F\)
['real']
  • length: 1
  • time: -2
  • mass: 1
force 21
2321 variable \omega
\(\omega\)
['real']
  • time: -1
angular frequency 21
1356 variable k
\(k\)
real
  • time: -2
  • mass: 1
linear stiffness, aka spring constant 7
5156 variable m
\(m\)
['real']
  • mass: 1
mass 57
4183 variable F_{\rm{spring}}
\(F_{\rm{spring}}\)
real
  • length: 1
  • time: -2
  • mass: 1
force of a spring 1
9140 variable a
\(a\)
['real']
  • length: 1
  • time: -2
acceleration 31
MESSAGE: