Applying the principle of homogeneity of dimensions, determine which one is correct, where \( T \) is time period, \( G \) is gravitational constant, \( M \) is mass, \( r \) is radius of orbit.
\( T^2 = 4\pi^2 r^3 \)
\( T^2 = \dfrac{4\pi^2 r}{GM^2} \)
\( T^2 = \dfrac{4\pi^2 r^2}{GM} \)
\( T^2 = \dfrac{4\pi^2 r^3}{GM} \)
Consider two physical quantities \( A \) and \( B \) related to each other as \( E = \dfrac{B - x^2}{A t} \) where \( E, x \) and \( t \) have dimensions of energy, length and time respectively. The dimension of \( AB \) is:
The de-Broglie wavelength associated with a particle of mass \(m\) and energy \(E\) is \( \dfrac{h}{\sqrt{2 m E}} \). The dimensional formula for Planck's constant is:
Match List-I with List-IIChoose the correct answer from the options given below:
The dimensional formula of latent heat is:
Match List-I with List-II.
Choose the correct answer from the options given below:
The angle between vector \( \vec{Q} \) and the resultant of \( (2\vec{Q} + 2\vec{P}) \) and \( (2\vec{Q} - 2\vec{P}) \) is:
The equation of state of real gas is given by \(\displaystyle \left( p + \frac{a}{V^{2}} \right)(V-b)=RT \), where \( P \), \( V \) and \( T \) are pressure, volume and temperature respectively and \( R \) is the universal gas constant. The dimensions of \(\displaystyle \frac{a}{b^{2}} \) is similar to that of:
Three vectors \( \overrightarrow{OP} \), \( \overrightarrow{OQ} \) and \( \overrightarrow{OR} \) each of magnitude \( A \) are acting as shown in figure. The resultant of the three vectors is \( A\sqrt{x} \). The value of \( x \) is _______.
For three vectors \( \vec{A} = (-x\hat{i} - 6\hat{j} - 2\hat{k}) \), \( \vec{B} = (-\hat{i} + 4\hat{j} + 3\hat{k}) \) and \( \vec{C} = (-8\hat{i} - \hat{j} + 3\hat{k}) \), if \( \vec{A}\cdot(\vec{B}\times\vec{C}) = 0 \), then value of \( x \) is ________.
There are \( 100 \) divisions on the circular scale of a screw gauge of pitch \( 1 \, \text{mm} \). With no measuring quantity in between the jaws, the zero of the circular scale lies \( 5 \) divisions below the reference line. The diameter of a wire is then measured using this screw gauge. It is found that \( 4 \) linear scale divisions are clearly visible while \( 60 \) divisions on circular scale coincide with the reference line. The diameter of the wire is:
The dimensional formula of angular impulse is:
The equation of stationary wave is: \( y = 2a \sin\!\left( \dfrac{2\pi n t}{\lambda} \right)\cos\!\left( \dfrac{2\pi x}{\lambda} \right) \). Which of the following is NOT correct:
If mass is written as \( m = k c^{P} G^{-1/2} h^{1/2} \) then the value of \( P \) will be: (Constants have their usual meaning with \( k \) a dimensionless constant).
If \( G \) be the gravitational constant and \( u \) be the energy density then which of the following quantity have the dimensions as that of the \( \sqrt{uG} \):
The measured value of the length of a simple pendulum is \( 20 \, \text{cm} \) with \( 2 \, \text{mm} \) accuracy. The time for \( 50 \) oscillations was measured to be \( 40 \, \text{seconds} \) with \( 1 \, \text{seconds} \) resolution. From these measurements, the accuracy in the measurement of acceleration due to gravity is \( N\% \). The value of \( N \) is:
In an expression \( a \times 10^{b} \).
If \( \varepsilon_0 \) is the permittivity of free space and \( E \) is the electric field, then \( \varepsilon_0 \) \(E^{2} \) has the dimensions:
