Part 1The given formula arises in the technical or scientific area of study shown. Solve for the indicated letter.aequals=StartFraction 2 mg Over Upper M plus 2 m EndFraction2mgM+2m for M (pulleys) Part 1Mequals=[input]2 m StartFraction left parenthesis g minus a right parenthesis Over a EndFraction2m(g − a)a

Question

Part 1The given formula arises in the technical or scientific area of study shown. Solve for the indicated letter.aequals=StartFraction 2 mg Over Upper M plus 2 m EndFraction2mgM+2m for M​ (pulleys) Part 1Mequals=[input]2 m StartFraction left parenthesis g  minus a right parenthesis Over a EndFraction2m(g − a)a

BlackTom AI · Accepted Answer

Start by isolating M from the equation a = (2 m g)/(M + 2 m) using basic algebra (multiplication, distribution, factoring, division).

1) Relevant knowledge points and concepts
- Algebraic manipulation to solve for a specific variable: multiply both sides by an expression containing the unknown, distribute, collect like terms, factor common factors, then divide.
- Verification by back-substitution to confirm the solution satisfies the original equation.

2) Step-by-step reasoning and algebra
Begin with the given relation:
a = (2 m g) / (M + 2 m)

Multiply both sides by (M + 2 m) to remove the denominator:
a(M + 2 m) = 2 m g

Distribute a on the left:
aM + 2 a m = 2 m g

Isolate the aM term by subtracting 2 a m from both sides:
aM = 2 m g − 2 a m

Factor 2 m on the right-hand side:
aM = 2 m (g − a)

Finally divide both sides by a to solve for M:
M = [2 m (g − a)] / a

3) Intermediate steps (explicit)
a(M + 2m) = 2mg
=> aM + 2am = 2mg
=> aM = 2mg − 2am
=> aM = 2m(g − a)
=> M = 2m(g − a)/a

4) Why this answer is correct
Algebraic operations are reversible and valid (multiplication, subtraction, factoring, division by a nonzero a). The solution isolates M explicitly. Back-substitution verifies it: substitute M into denominator:
M + 2m = 2m(g − a)/a + 2m = 2m[(g − a)/a + 1] = 2m[g/a], so (2mg)/(M + 2m) = (2mg)/(2m[g/a]) = a, confirming correctness.

5) Why other common incorrect forms are wrong
- M = (2mg/a) − 2m is algebraically equivalent to the correct form; however writing M = (2mg/a) + 2m (wrong sign) would not satisfy the original equation.
- Errors often come from forgetting to distribute a or from sign mistakes when moving terms; those give expressions that do not simplify back to a when substituted.

6) Conclusion
The solved expression M = 2m(g − a)/a matches the provided answer exactly.

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