A force $F$ is given by $F = at + bt^2$ where $t$ is time. The dimensions of $a$ and $b$ are:
If the dimensions of length are written as $E^{\alpha} v^{\beta} F^{\gamma}$ where $E, v, F$ are energy, velocity, and force respectively, then:
In the equation $\left(P + \dfrac{a}{V^2}\right)(V - b) = RT$ (van der Waals equation), the dimensions of $a/b$ are:
The dimensions of $\dfrac{e^2}{4\pi\varepsilon_0 hc}$ (the fine-structure constant) are:
The mean time period of a simple pendulum from $4$ trials is $2.51, 2.49, 2.50, 2.52$ s. The percentage error in the mean is approximately:
The wavelength $\lambda$ of matter wave depends on momentum $p$ and Planck's constant $h$ as $\lambda \propto h^x p^y$. The values are:
The frequency of vibration $\nu$ of a string under tension depends on length $L$, tension $T$, and mass per unit length $\mu$. From dimensional analysis:
A student measures $g$ using a simple pendulum and obtains a percentage error of $3\%$ for $L$ and $2\%$ for $T$. The maximum percentage error in $g$ is:
A wire has resistance $R = V/I$ where $V = (50 \pm 1)$ V and $I = (10 \pm 0.2)$ A. The percentage error in $R$ is:
The Reynolds number $Re = \dfrac{\rho v d}{\eta}$ where $\rho$ = density, $v$ = velocity, $d$ = length, $\eta$ = viscosity. Its dimensions are:
A physical quantity $Q$ is related to length $L$ as $Q = AL + B/L$. If the dimensions of $Q$ are $[ML^2T^{-2}]$, and the dimensions of $A$ are $[M^aL^bT^c]$, find $a + b + c$.
A quantity $X = \dfrac{1}{2}LI^2$ has dimensions of energy. If $L$ is inductance and $I$ is current, find the dimensions of inductance $L$ in the form $[ML^xT^yA^z]$. Give $x + y + z$.
In an experiment, a steel ball of diameter $0.50$ cm is dropped. The screw gauge has LC = $0.001$ cm and a positive zero error of $0.005$ cm. The student records the diameter as $0.495$ cm. The corrected diameter in mm (×100) is the integer:
Assertion (A): Dimensional analysis can be used to derive the formula $E = mc^2$. Reason (R): All dimensionally correct equations can be derived using dimensional analysis alone.
Assertion (A): In the formula $y = a \sin(\omega t + \phi)$, both $\omega t$ and $\phi$ must be dimensionless. Reason (R): Arguments of trigonometric functions must always be dimensionless angles.
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