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Zorluk: KolayPhysical Quantities, Units and Dimensions

The physical quantity work is defined as the product of force and displacement. What are the dimensional exponents aa, bb, and cc for mass, length, and time respectively in the dimensional formula for work, [MaLbTc][M^a L^b T^c]?

  1. 1,2,−21, 2, -2Cevap
  2. B
    1,1,−21, 1, -2
  3. C
    1,2,−11, 2, -1
  4. D
    1,−3,01, -3, 0

Cevap

The dimensional exponents for mass, length, and time in the expression for work are a=1a = 1, b=2b = 2, and c=−2c = -2.
Work is calculated as force multiplied by displacement. Since force has dimensions [MLT−2][M L T^{-2}] and displacement has dimension [L][L], the resulting dimensional formula for work is [M1L2T−2][M^1 L^2 T^{-2}]. The exponents of MM, LL, and TT are therefore 11, 22, and −2-2 respectively.

Adım Adım Çözüm

1
Express the dimensions of force using base quantities
\text{Force} = \text{mass} \times \text{acceleration} = [M] \times [L T^{-2}] = [M L T^{-2}]
Acceleration has the dimensions of length divided by time squared.
2
Multiply force dimensions by displacement dimension to find the dimensions of work
\text{Work} = \text{Force} \times \text{Displacement} = [M L T^{-2}] \times [L] = [M^1 L^2 T^{-2}]
Displacement is a measure of length, contributing an additional factor of [L][L].
3
Extract the exponents aa, bb, and cc
a = 1, b = 2, c = -2
Comparing [MaLbTc][M^a L^b T^c] to [M1L2T−2][M^1 L^2 T^{-2}] gives the values of aa, bb, and cc.

Anahtar Kavram

Dimensional analysis of work
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Physical Quantities, Units and Dimensions

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