The angle \(1’\) (minute of arc) in radian is nearly equal to,
[2020- Covid]
Convert \(1'\) to degrees, then to radians. \(1' = (1/60)^\circ = (1/60) \times (\pi/180)\text{ rad} \approx 2.91 \times 10^{-4}\text{ rad}\)
The unit of thermal conductivity is
[2019]
Thermal conductivity \(k\) relates heat flux to temperature gradient. So, \(k\) has units of \(W/(m \cdot K) = W m^{-1} K^{-1}\).
The damping force on an oscillator is directly proportional to the velocity. The units of the constant of proportionality are:
[2012 pre]
Damping force \(F_d = -bv\). The constant \(b\) has dimensions \([F_d]/[v] = (MLT^{-2})/(LT^{-1}) = MT^{-1}\), which is \(kg s^{-1}\).
The density of a material in CGS system of units is \(4\text{ g/cm}^3\). In a system of units in which unit of length is \(10\text{ cm}\) and unit of mass is \(100\text{ g}\), the value of density of material will be:
[2011 Mains]
Given \(\rho = 4\text{ g/cm}^3\). New mass unit \(M' = 100\text{ g}\) and new length unit \(L' = 10\text{ cm}\). \(\rho = 4 \frac{1/100 M'}{(1/10 L')^3} = 4 \frac{1/100}{1/1000} \frac{M'}{L'^3} = 40 \frac{M'}{L'^3}\).
If \(x = at + bt^2\), where \(x\) is the distance travelled by the body in kilometres while \(t\) is the time in seconds, then the units of \(b\) is:
[1989]
By homogeneity, \([bt^2] = [x]\). So \([b] = [x]/[t^2] = km/s^2\).
Plane angle and solid angle have:
[2022]
Plane angle (radian) and solid angle (steradian) are dimensionless quantities (ratios of lengths or areas) but possess units.
The dimension \([ML^{-2}T^{-2}A^2]\) belong to the:
[2022]
The dimension of magnetic permeability is \(MLT^{-2}A^{-2}\). The given dimension \(ML^{-2}T^{-2}A^2\) does not match a standard physical quantity listed. Assuming a typo and it refers to magnetic permeability.
If \(E\) and \(G\) respectively denote energy and gravitational constant, then \(\frac{E}{G}\) has the dimensions of:
[2021]
Energy \([E] = ML^2T^{-2}\) and Gravitational Constant \([G] = M^{-1}L^3T^{-2}\). So \([E/G] = M^2L^{-1}T^0\).
Dimensions of stress are:
[2020]
Stress is Force per unit Area. \([\text{Stress}] = [F]/[A] = (MLT^{-2})/L^2 = ML^{-1}T^{-2}\).
The dimensions of \((\mu_0\epsilon_0)^{-1/2}\) are:
[2012 Mains]
The speed of light \(c = 1/\sqrt{\mu_0\epsilon_0}\). Therefore, \((\mu_0\epsilon_0)^{-1/2} = c\). The dimensions of speed are \([LT^{-1}]\).