## 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
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:
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
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:
$$-\frac{k}{m} x = -A \omega^2 \cos(\omega t)$$
$$- \frac{k x}{pdg_{5156}} = - A pdg_{2321}^{2} \cos{\left(pdg_{2321} pdg_{9491} \right)}$$
input diff is d**2*(-pdg4037 + x)/dt**2 diff is -k*x/pdg5156 + pdg2321**2*pdg9885*cos(pdg1467*pdg2321) diff is -A*pdg2321**2*cos(pdg2321*pdg9491) + 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:
$$-\frac{k}{m} x = -A \omega^2 \cos(\omega t)$$
$$- \frac{k x}{pdg_{5156}} = - A pdg_{2321}^{2} \cos{\left(pdg_{2321} pdg_{9491} \right)}$$
1. 2148049269; locally 7745098:
$$-\frac{k}{m} A \cos(\omega t) = -A \omega^2 \cos(\omega t)$$
$$- \frac{A k \cos{\left(pdg_{2321} pdg_{9491} \right)}}{pdg_{5156}} = - A pdg_{2321}^{2} \cos{\left(pdg_{2321} pdg_{9491} \right)}$$
LHS diff is k*(A*cos(pdg2321*pdg9491) - x)/pdg5156 RHS diff is 0 5415824175:
1772973171:
2148049269:
5415824175:
1772973171:
2148049269:
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:
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:
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:
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:
12 multiply both sides by
1. 2148049269; locally 7745098:
$$-\frac{k}{m} A \cos(\omega t) = -A \omega^2 \cos(\omega t)$$
$$- \frac{A k \cos{\left(pdg_{2321} pdg_{9491} \right)}}{pdg_{5156}} = - A pdg_{2321}^{2} \cos{\left(pdg_{2321} pdg_{9491} \right)}$$
1. 7473576008:
$$\frac{-1}{A \cos(\omega t)}$$
$$pdg_{1467}$$
1. 1931103031; locally 8360924:
$$\frac{k}{m} = \omega^2$$
$$\frac{pdg_{1356}}{pdg_{5156}} = pdg_{2321}^{2}$$
LHS diff is -(A*k*pdg1467*cos(pdg2321*pdg9491) + pdg1356)/pdg5156 RHS diff is -pdg2321**2*(A*pdg1467*cos(pdg2321*pdg9491) + 1) 2148049269:
1931103031:
2148049269:
1931103031:
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:
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:
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:
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

symbol ID category latex scope dimension name value Used in derivations references
9491 variable T
$$T$$
['real']
• time: 1
period 20
1356 variable k
$$k$$
real
• mass: 1
• time: -2
linear stiffness, aka spring constant
7
4183 variable F_{\rm{spring}}
$$F_{\rm{spring}}$$
real
• length: 1
• mass: 1
• time: -2
force of a spring
1
1467 variable t
$$t$$
['real']
• time: 1
time
121
9885 variable A
$$A$$
real dimensionless amplitude
4
4037 variable x
$$x$$
['real']
• length: 1
position
53
9140 variable a
$$a$$
['real']
• length: 1
• time: -2
acceleration 31
4202 variable F
$$F$$
['real']
• length: 1
• mass: 1
• time: -2
force
21
2321 variable \omega
$$\omega$$
['real']
• time: -1
angular frequency
26
5156 variable m
$$m$$
['real']
• mass: 1
mass
69
MESSAGE:
• local variable 'all_df' referenced before assignment