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Physics · UNITS AND MEASUREMENT
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Physics · UNITS AND MEASUREMENT
Plane angle \( d\theta \) and solid angle \( d\Omega \) are defined as the ratio of arc length to radius and intercepted area to square of radius, respectively. Which of the following statements is correct regarding their dimensions and units?
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What is defined as a certain basic, arbitrarily chosen, internationally accepted reference standard used in the measurement of any physical quantity?
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Consider the following statements regarding the method of dimensional analysis: Statement I: If an equation is dimensionally correct, it is guaranteed to be physically correct. Statement II: The dimensionless constant in a physical relation (such as \( 2\pi \) in \( T = 2\pi\sqrt{l/g} \)) cannot be determined using dimensional analysis alone. Statement III: A formula involving the sum of two terms having different dimensions, such as \( K = \frac{1}{2}mv^2 + ma \), is dimensionally invalid according to the principle of homogeneity. Which of the above statements are correct?
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A substance of mass \( 5.74 \text{ g} \) occupies a volume of \( 1.2 \text{ cm}^3 \). Express its density to the correct number of significant figures, and identify the underlying rule.
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As shown in the figure, plane angle \( d\theta \) is defined as the ratio of the length of arc \( ds \) to the radius \( r \). What is the SI unit for this plane angle?
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Which of the following sets of physical quantities have different dimensional formulas?
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The mole is the SI unit of amount of substance. One mole contains exactly how many elementary entities?
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The position \( x \) of a particle at time \( t \) is given by \( x = \frac{v_0}{A}(1 - e^{-A t}) \), where \( v_0 \) is constant and \( A > 0 \). The dimensions of \( v_0 \) and \( A \) respectively are:
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In an experiment, four quantities \( a, b, c \) and \( d \) are measured with percentage errors \( 1\% \), \( 2\% \), \( 3\% \) and \( 4\% \) respectively. Quantity \( P \) is calculated as \( P = \frac{a^3 b^2}{\sqrt{c} d} \). The maximum percentage error in \( P \) is:
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In a wave equation, the displacement \( y \) of a wave as a function of time \( t \) and distance \( x \) is given by \( y = A e^{-\alpha t} \sin(\beta x - \gamma t) \). The dimensional formula of \( \frac{\gamma}{\beta} \) is equivalent to which of the following physical quantities?
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Two masses are measured using a balance with an absolute error of \( \pm 0.01\text{ g} \). The first mass is \( m_1 = 1.00\text{ g} \) and the second mass is \( m_2 = 10.00\text{ g} \). The ratio of the relative error in the measurement of \( m_1 \) to that of \( m_2 \) is:
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The mean density of the Sun, calculated using its mass \( M = 2.0 \times 10^{30}\text{ kg} \) and radius \( R = 7.0 \times 10^8\text{ m} \), is closest to which order of density range?
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Applying the standard rules of rounding off to three significant figures, the numbers \( 2.745 \) and \( 2.735 \) become respectively:
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The number of significant figures in \( 123000\text{ mm} \) is:
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The base units for length, mass, and time respectively in the CGS system of units are:
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Which of the following limitations is inherent to the method of dimensional analysis?
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In the FPS (or British) system of units, what were the base units for length, mass, and time, respectively?
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According to the general rule for multi-step computations, intermediate results should be calculated to:
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The SI unit of energy is \( \text{J} = \text{kg m}^2 \text{ s}^{-2} \). A student proposes four different expressions for a physical quantity \( Q \) which has the dimensions of energy: (a) \( Q_1 = m^2 v^3 t^{-1} \) (b) \( Q_2 = \frac{1}{2} m v^2 \) (c) \( Q_3 = m a x \) (d) \( Q_4 = m v t \) where \( m \) is mass, \( v \) is speed, \( a \) is acceleration, \( x \) is position, and \( t \) is time. Which of the following options lists ALL expressions that are dimensionally CONSISTENT with energy?
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In the equation \( P = \frac{a - t^2}{b x} \), where \( P \) is pressure, \( x \) is distance, and \( t \) is time, the dimensions of \( \frac{a}{b} \) are:
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What is the volume of a cube of side \( 1\text{ cm} \) expressed in \( \text{m}^3 \)?
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What are the dimensions of a physical quantity defined as?
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