A student designed a new system of measurement in which momentum (P), energy (E), and length (L) are taken as fundamental quantities. The dimension of acceleration in this system i... A student designed a new system of measurement in which momentum (P), energy (E), and length (L) are taken as fundamental quantities. The dimension of acceleration in this system is given as [P^α E^β L^γ], then |α| + |β| + |γ| is?
Understand the Problem
The question is asking for the values of α, β, and γ in the dimension of acceleration expressed in terms of momentum (P), energy (E), and length (L). This involves deriving the dimensions and requires knowledge of dimensional analysis in physics.
Answer
$5$
Answer for screen readers
The value of $|\alpha| + |\beta| + |\gamma|$ is $5$.
Steps to Solve
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Identify the dimensions of the quantities
- Momentum ($P$) has dimension $[P] = [M][L][T]^{-1}$.
- Energy ($E$) has dimension $[E] = [M][L]^2[T]^{-2}$.
- Length ($L$) has dimension $[L] = [L]$.
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Write the dimension of acceleration
- Acceleration ($a$) has dimension $[a] = [L][T]^{-2}$.
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Express acceleration in terms of $P$, $E$, and $L$
- According to the problem statement: $$ [a] = [P]^{\alpha}[E]^{\beta}[L]^{\gamma} $$
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Substitute the dimensions into the equation
- Replace $[P]$, $[E]$, and $[L]$ with their dimensions: $$ [L][T]^{-2} = ([M][L][T]^{-1})^{\alpha}([M][L]^2[T]^{-2})^{\beta}([L])^{\gamma} $$
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Expand the right side
- Expanding gives: $$ [L][T]^{-2} = [M]^{\alpha + \beta} [L]^{\alpha + 2\beta + \gamma}[T]^{-\alpha - 2\beta} $$
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Set up equations for each dimension
- Equate the dimensions:
- For mass $M$: $$ \alpha + \beta = 0 $$
- For length $L$: $$ \alpha + 2\beta + \gamma = 1 $$
- For time $T$: $$ -\alpha - 2\beta = -2 $$
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Solve the equations
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From $\alpha + \beta = 0$, we find $\beta = -\alpha$.
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Substituting $\beta = -\alpha$ into the other equations:
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Substitute into $ -\alpha - 2(-\alpha) = -2$: $$ -\alpha + 2\alpha = -2 $$ $$ \alpha = -2 $$
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Use $\alpha = -2$ to find $\beta$: $$ \beta = -(-2) = 2 $$
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Substitute into $\alpha + 2\beta + \gamma = 1$: $$ -2 + 2(2) + \gamma = 1 $$ $$ -2 + 4 + \gamma = 1 $$ $$ \gamma = -1 $$
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Find the sum of the absolute values
- Calculate $|\alpha| + |\beta| + |\gamma|$: $$ |\alpha| + |\beta| + |\gamma| = |{-2}| + |{2}| + |{-1}| = 2 + 2 + 1 = 5 $$
The value of $|\alpha| + |\beta| + |\gamma|$ is $5$.
More Information
In this calculation, we found the coefficients related to the dimensional analysis of acceleration when expressed in terms of momentum, energy, and length. The values of $\alpha$, $\beta$, and $\gamma$ reveal the interrelation between these physical quantities and their respective dimensions.
Tips
- Confusing the dimensions of energy and momentum can lead to incorrect results. Always remember that momentum involves mass times velocity while energy involves mass times velocity squared.
- Neglecting to carry through the signs when solving equations can lead to miscalculations in $\alpha$, $\beta$, and $\gamma$.
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