Units and Measurement

The force is given in terms of time t and displacement x by the equation
F = A cos Bx + C sin Dt
The dimensional formula of AD/B is :

The work done by a gas molecule in an isolated system is given by $W=\alpha {\beta }^2{e}^{-\frac{x^2}{\alpha kT}}$ <!-- [if gte mso 9]><xml> </xml><![endif]-->, where x is the displacement, k is the Boltzmann constant and T is the temperature. a and b are constants. Then the dimensions of b will be:

In a typical combustion engine the work done by a gas molecule is given by W = ${\alpha }^2\beta e^{-\frac{\beta x^2}{kT}},$ where x is the displacement, k is the Boltzmann constant and T is the temperature. If and are constants, dimensions of will be

The period of oscillation of a simple pendulum is $T=2\pi \sqrt[]{\frac{L}{g}}.$ <!-- [if gte mso 9]><xml> </xml><![endif]--> Measured value of ‘L’ is 1.0m from meter scale having a minimum division of 1mm and time of one-complete oscillation is 1.95s measured from stopwatch of 0.01s resolution. The percentage error in the determination of ‘g’ will be:

The pitch of the screw gauge is 1mm and there are 100 divisions on the circular scale. When nothing is put in between the jaws, the zero of the circular scale lies 8 divisions below the reference line. When a wire is placed between the jaws, the first linear scale division is clearly visible while 72^nd division on circular scale coincides with the reference line, The radius of the wire is:

If e is the electronic charge, c is the speed of light in free space and h is Planck’s constant, the quantity $\frac{1}{4\pi {\epsilon }_0}\frac{{\left|e\right|}^2}{hc}$ has dimensions of

A force defined by $F=\alpha t^2+\beta t$ acts on a particle at a given time $t$ . The factor which is dimensionless, if $\alpha$ and $\beta$  are constants, is:

In a vernier callipers, $\left(N+1\right)$ divisions of vernier scale coincide with $N$  divisions of main scale. If 1 MSD represents 0.1 mm, the vernier constant (in cm) is:

A wire of length 'l' and resistance $100\Omega$  is divided into 10 equal parts. The first 5 parts are connected in series while the next 5 parts are connected in parallel. The two combinations are again connected in series. The resistance of this final combination is:

In an experiment to determine the Young’s modulus of wire of a length exactly 1m, the extension in the length of the wire is measured as 0.4mm with an uncertainty of $\pm$ <!-- [if gte mso 9]><xml> </xml><![endif]-->0.02mm when a load of 1kg is applied. The diameter of the wire is measured as 0.4mm with an uncertainty of $\pm$ <!-- [if gte mso 9]><xml> </xml><![endif]-->0.02mm. The error in the measurement of Young’s modulus $\left(\Delta Y\right)$ <!-- [if gte mso 9]><xml> </xml><![endif]--> is found to be x × 10¹⁰Nm^-2. The value of x is _________.

(take g = 10 ms^-2)