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Convert 2.5 g/mm2 to kg/m2 and check the dimensional correctness of T = 2pisqrt(m/(F/x)).

Physics

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2 parts
a

2.5×103 kgm−22.5 \times 10^3 \text{ kgm}^{-2}

b

dimensionally correct\text{dimensionally correct}

Step-by-step solution

9 steps

Here are the solutions to the questions from the image:

Part a· 3 steps

Convert 2.5 gmm−22.5 \text{ gmm}^{-2} to kgm−2\text{kgm}^{-2}.

  1. Convert grams (g) to kilograms (kg).

    We know that 1 g=10−3 kg1 \text{ g} = 10^{-3} \text{ kg}.

  2. Convert square millimeters (mm2\text{mm}^2) to square meters (m2\text{m}^2).

    We know that 1 mm=10−3 m1 \text{ mm} = 10^{-3} \text{ m}.
    Therefore, 1 mm−2=(10−3 m)−2=106 m−21 \text{ mm}^{-2} = (10^{-3} \text{ m})^{-2} = 10^6 \text{ m}^{-2}.

  3. Substitute the conversions into the given value.

    2.5 gmm−2=2.5×(10−3 kg)×(106 m−2)2.5 \text{ gmm}^{-2} = 2.5 \times (10^{-3} \text{ kg}) \times (10^6 \text{ m}^{-2}) 2.5 gmm−2=2.5×10(−3+6) kgm−22.5 \text{ gmm}^{-2} = 2.5 \times 10^{(-3+6)} \text{ kgm}^{-2} 2.5 gmm−2=2.5×103 kgm−22.5 \text{ gmm}^{-2} = 2.5 \times 10^3 \text{ kgm}^{-2}

    The converted value is 2.5×103 kgm−2\boxed{2.5 \times 10^3 \text{ kgm}^{-2}}.

Part b

dimensionally correct\text{dimensionally correct}

Determine if the formula T=2πmF/xT = 2\pi \sqrt{\frac{m}{F/x}} is dimensionally correct. Where TT is time period, mm is mass, FF is force and xx is distance.

  1. Determine the dimension of the Left Hand Side (LHS).

    The LHS is TT, which represents time period.
    Dimension of LHS = [T][T]

  2. Determine the dimensions of the terms in the Right Hand Side (RHS).

    2π2\pi is a dimensionless constant.
    Dimension of mass (mm) = [M][M]
    Dimension of force (FF) = [M][L][T−2][M][L][T^{-2}]
    Dimension of distance (xx) = [L][L]

  3. Calculate the dimension of the term Fx\frac{F}{x}.

    Dimension of Fx=[M][L][T−2][L]=[M][T−2]\frac{F}{x} = \frac{[M][L][T^{-2}]}{[L]} = [M][T^{-2}]

  4. Calculate the dimension of the term mF/x\frac{m}{F/x}.

    Dimension of mF/x=[M][M][T−2]=1[T−2]=[T2]\frac{m}{F/x} = \frac{[M]}{[M][T^{-2}]} = \frac{1}{[T^{-2}]} = [T^2]

  5. Calculate the dimension of the square root term.

    Dimension of mF/x=[T2]=[T]\sqrt{\frac{m}{F/x}} = \sqrt{[T^2]} = [T]

  6. Compare the dimensions of LHS and RHS.

    Dimension of LHS = [T][T]
    Dimension of RHS = Dimension of 2π×2\pi \times Dimension of mF/x=1×[T]=[T]\sqrt{\frac{m}{F/x}} = 1 \times [T] = [T]
    Since the dimension of LHS equals the dimension of RHS, the formula is dimensionally correct.
    The formula is dimensionally correct\boxed{\text{dimensionally correct}}.

Part c

State the conditions for two vectors to be equal. Give any four examples of the term "magnitude" as applied to vector and scalar quantities.

Conditions for two vectors to be equal:

  1. They must have the same magnitude.
  2. They must have the same direction.
  3. They must represent the same physical quantity.

Examples of quantities that have magnitude:

  • Mass (Scalar): e.g., 5 kg5 \text{ kg}
  • Velocity (Vector): e.g., 20 ms−120 \text{ ms}^{-1} North
  • Temperature (Scalar): e.g., 30∘C30^\circ \text{C}
  • Force (
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