Physics Test-20

We know that,

\(\frac{{dN}}{{dt}}=\lambda {N}\)

Now,

\(\frac{d N_{0}}{d t}=\lambda N_{0} ; \frac{d N_{t}}{d t}=\lambda N_{t}\)

\(9750=\lambda {N}_{0} ; 975=\lambda {N}_{{t}}\)

\(\frac{{N}_{0}}{{~N}_{{t}}}=\frac{9750}{975}=\frac{10}{1}\)

\(\Rightarrow {N}_{0}=10 {~N}_{{t}}\)

We know that,

\(N_{t}=N_{0} e^{-\lambda t} \)

\(\frac{N_{t}}{N_{0}}=e^{-\lambda t}\)

\(\frac{1}{10}=e^{-\lambda .5} \)

\(10^{-1}=e^{-5 \lambda}\)

Taking log on both sides,

\(-1=-5 \lambda \times \frac{1}{2.303} \)

\(\lambda=\frac{1}{5} \times 2.303=0.461 \) per minute

Given:

Mass of a planet (Mp) = \(\frac{1}{4}\)th of the mass of the earth \((M_e)\)

The radius of a planet (rp) = \(\frac{1}{9}\)th the radius of the earth \((r_e)\)

Escape velocity from the earth (Ve) = 11 km/s

The escape velocity on the earth is given by:

\( V_{e}=\sqrt{\frac{2 G M_{e}}{r_{e}}}\).....(1)

The escape velocity on the planet is given by:

\( V_{p}=\sqrt{\frac{2 G M_{p}}{r_{p}}}\)

\(=\sqrt{\frac{2 G \frac{M_{e}}{4}}{\frac{r^{e}}{9}}} \)

\(=\sqrt{\frac{2\left(9 G M_{e}\right)}{4 e_{e}}}\)

\(=\frac{3}{2} \sqrt{\frac{2 G M_{e}}{r_{e}}}\).....(2)

On dividing equation (1) and (2), we get:

\( \frac{V_{e}}{V_{p}}=\frac{2}{3} \)

\(\Rightarrow V_{p}=\frac{3}{2} V_{e}\)

\(=\frac{33}{2} km / s\)

When the widths of the two slits are increased, the fringes become brighter. However, the width of each slit should be considerably smaller than the separation between the slits. When the slits become so wide that this condition is not satisfied, the interference pattern disappears i.e., The fringes become less distinct.

The partial pressure of gas A \(=\frac{P_{1} V_{1}}{V_{2}}\)

\(=\frac{600  \times 1000 }{2000}\)

\(=300\) torr

Similarly partial pressure of the gas B

\(=\frac{1000  \times 500 }{2000 }\)

\(=250\) torr

So total pressure of the mixture \(=300\) torr \(+250\) torr

\(=550\) torr

The magnetic field B inside the toroid is constant in magnitude for the ideal toroid of closely wound turns. The magnetic field in the open space inside (point P) and exterior to the toroid (point Q) is zero.

The magnetic field \(B\) inside the toroid is given as,

\(B=\frac{\mu_{o} N I}{2 \pi r}\)

Where, \(N =\) number of turns, \(I =\) current, and \(r =\) average radius of the toroid, If \(N\) and \(I\) are constant, then,

\(B \propto \frac{1}{r}\)

\(\Rightarrow B \propto r^{-1}\)

Given that, the temperature of freezer,

\(T_{2}=\) \(-13^{\circ} C\)

\(\Rightarrow \quad T_{2}=-13+273=260 K\)

Coefficient of performance, \(\beta=5\)

The coefficient of performance is defined as,

\(\beta=\frac{T_{2}}{T_{1}-T_{2}}\)

\(=5=\frac{260}{T_{1}-260}\)

\(= T_{1}-260=\frac{260}{5}\)

\(=T_{1}-260=52\)

\(\operatorname{T}_{1}=(52+260) K =312 K\)

\(T_{1}=(312-273)^{\circ} C\)

\(T_{1}=39^{\circ} C\)

According to Wien's law, the wavelength corresponding to the maximum energy is inversely proportional to the absolute temperature (T).

Wien's displacement law states that the wavelength corresponding to maximum energy is inversely proportional to the temperature of the body in Kelvin. It can also be said that the black-body radiation curve for different temperatures will peak at different wavelengths that are inversely proportional to the temperature.

Crystalline solids are those in which atoms are arranged in an orderly fashion. They have directional properties and therefore are called anisotropic substances.

Active power and apparent power are respectively measured inkW and kVA.

Power factor of an AC circuit is defined as the ratio of the active power consumed by a circuit to the apparent power consumed by the same circuit.The power which is actually consumed or utilized in an AC Circuit is called active power. It is measured in kW or MW.The product of the root-mean-square value of voltage and current is known as apparent power. It is measured in kVA or MVA.