Physics Test-8

Given that:

The satellite is shifted from an orbit of radius 3R to 5R.

Let TE₁ and TE₂ be the energy of the satellite in the orbit of radius 3R and 5R respectively.

\(T E_{1}=-\frac{G m M}{2(3 R)}\) and \(T E_{2}=-\frac{G m M}{2(5 R)}\)

Change in energy, ΔTE = TE₂ - TE₁

\(\Rightarrow \Delta T E=-\frac{G m M}{2(5 R)}-\left(-\frac{G m M}{2(3 R)}\right)\)

\(\Rightarrow \Delta T E=\frac{1}{15}\left(\frac{G m M}{R}\right)\)

Percentage increase \(=\frac{\Delta T E}{T E_{1}} \times 100\)

\(= \frac{\frac{1}{15}\left(\frac{G m M}{R}\right)}{\frac{G m M}{2(3 R)}} \times 100\)

\(= \frac{\frac{1}{15}}{\frac{1}{6}} \times 100\)

\(= 40 \%\)

Power is defined as the work done per unit time.

• Dimensions of work done = Force × displacement = [M¹L¹T^-2] × [L¹] = [M¹L²T^-2]
• Dimensions of time = [T1]
• Thus, the dimensions of power = [M¹L²T^-2] × [T¹]^-1 = [M¹L²T^-3]

​Dimensions of speed = [M⁰L¹T^-1]

⇒ Muscular strength × Speed = Power

⇒ Muscular strength × [M⁰L¹T^-1] = [M¹L²T^-3]

⇒ Muscular strength = [M¹L¹T^-2]

The given sample is diamagnet and hence it has a magnetic susceptibility χ < 0 i.e. negative.

A change in temperature can only affect the random thermal motion of the atom.

• Since each atom of a diamagnetic substance has zero magnetic dipole moment, the temperature cannot affect the net magnetic moment in any way.
• Thus, diamagnetic substances are independent of temperature.
• Therefore, magnetic susceptibility remains constant irrespective of the temperature.

​So, option (B) depicts the correct dependence of magnetic susceptibility χ with temperature.

Here isothermal process is given,

Temperature \(( T )=\) constant

\(PV = n R T =\) constant

\(P _{1} V _{1}= P _{2} V _{2} \quad \ldots\) (1)

Where, \(P _{1}\) is initial pressure, \(P _{2}\) is final pressure, \(V _{1}\) is initial volume and \(V _{2}\) is final pressure.

Since, pressure is increased by \(10 \%\)

Final pressure \(\left(P_{2}\right)=P+P \times 10 \%=1.1 P\)

Use equation (1);

\(P _{1} V _{1}= P _{2} V _{2}\)

\(P \times V _{1}=1.1 P \times V _{2}\)

\(V _{2}= \frac{V _{1}}{1.1}=\frac{10 V_{1} }{11}\)

Decrease in volume \((\Delta V )= V _{1}- V _{2}= V_{1} -\frac{10 V_{1}}{11}= \frac{V_{1}}{11}\)

\(\%\) Decrease in volume \(=\frac{\Delta V}{V_{1}} \times 100=\frac{V_{1}}{11 V_{1}} \times 100=9.09 \approx 9 \%\)

Sodium, lithium, and magnesium are electropositive and they also lose electrons easily but their reactivity is much less than Caesium.

Cesium is the most electropositive element of all so it has the minimum ionization energy and so contains the maximum capacity to lose electrons.

Newton’s law of gravitation gives the force between two mass particles separated by some distance. So option (A) does not follow.

Newton’s first law of motion, an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. So option (B) does not follow.

According to Newton’s second law of motion:

Force \((F)=\frac{\Delta P}{\Delta t}\)

If there is no external force then, F = 0

Force \((F)=\frac{\Delta P}{\Delta t}=0\)

\(\frac{\Delta P}{\Delta t}=\frac{P_{1}-P_{2}}{\Delta t}=0\)

\(\left({P}_{1}-{P}_{2}\right)=0\)

• P₁ = P₂ Hence the initial momentum is equal to final momentum if there is no external force on the system. So this is the law of the conservation of linear momentum. So it explains the conservation of momentum.
• Newton’s third law of motion states that for every action, there is an equal and opposite reaction. It also explains the conservation of momentum.

When a thermodynamic system undergoes a physical change in such a way that its temperature remains constant, then the change is known as an isothermal process.

As we know that, the internal energy of the system is a function of temperature alone, so in the isothermal process, the change in internal energy is zero.

⇒ ΔQ = 0 + ΔW = ΔW

Therefore, the heat given to an ideal gas in isothermal conditions is used to do external work.

With ‘p’ for proton and 'n’ for neutron, the nuclear forces have strengths in n - n = p - p = p - n.

Nucleons in the nucleus of an atom are bonded together by a strong force of attraction called nuclear force. The nuclear force is short-ranged force that acts when nucleons are at distance in the range of fermi (1 fermi = 10 ^-15 m). It acts between proton-proton, proton - neutron, and neutron - neutron. 

In vacuum an electron of energy 10 keV hits tungeston target, then emitted radiation will be X-rays.

X-Rays: X-rays are powerful waves of electromagnetic energy.

• Most of them have a wavelength ranging from 0.01 to 10 nanometres, corresponding to frequencies in the range of 30 petahertz to 30 exahertz.
• The energies of X-Rays are in the range of 100 eV to 100 keV.
• X-rays are produced when high-velocity electrons collide with the metal plates, thereby giving the energy as the X-Rays and the electrons get absorbed by the metal plate.

Simple pendulum:

• When a point mass is attached to an inextensible string and suspended from fixed support then it is called a simple pendulum.
• The time period of a simple pendulum is defined as the time taken by the pendulum to finish one complete oscillation.

\(\Rightarrow T=2 \pi \sqrt{\frac{l}{g}}\)

• The above formula is only valid for small angular displacements.

Where, T = Time period of oscillation, l = length of the pendulum, and g = gravitational acceleration

• The amplitude of a simple pendulum is defined as the maximum distance traveled by the pendulum from the equilibrium position to one side.
• The length of a simple pendulum is defined as the distance between the point of suspension to the center of the bob.

According to the definition of the simple pendulum, the bob must be a point mass. So statement (B) is correct.

If the size of the bob will be more then more surface area will be exposed in the air and more resistance will act which will alter the time period of the pendulum. So statement (A) is correct.

Due to the large size, the net distance between the point of the hanging of the pendulum and the center of the bob will increase. This length is called an effective length.

This increase in the effective length of the pendulum will affect the time period. So, statement (C) is correct. So option (D) is correct.

Given: \(I_{1}=I_{2}=1, A_{1}=A_{2}=A\) and \(\rho_{2}=4 \rho_{1}\)

For case (i), \(\left(\rho_{1}=\rho\right)\)

\(\Rightarrow R_{1}=\frac{\rho l}{A}\) .....(i)

The current in the circuit,

\(\Rightarrow I_{1}=\frac{V}{R_{1}}\)

\(\Rightarrow I_{1}=\frac{V A}{\rho l}\) .....(ii)

For case (ii), \(\left(\rho_{2}=4 \rho\right)\)

\(\Rightarrow R_{2}=\frac{4 \rho l}{A}\) .....(iii)

The current in the circuit

\(\Rightarrow I_{2}=\frac{V}{R_{2}}\)

\(\Rightarrow I_{2}=\frac{V A}{4 \rho l}\) .....(iv)

By equation (ii) and equation (iv),

\(\Rightarrow I_{2}=\frac{I_{1}}{4}\)

Given f_c + f_m = 1020 kHz and f_c - f_m = 980 kHz

Where f_c = frequency of the carrier wave and f_m = frequency of the modulating signal

• We know that that (f_c + f_m) and (f_c - f_m) are the sidebands of a modulated signal.
• Therefore by adding the two sidebands,

⇒ (f_c + f_m) + (f_c - f_m) = (1020 + 980) kHz

⇒ 2f_c = 2000 kHz

⇒ f_c = 1000 kHz

⇒ f_c = 1 MHz

Dielectrics are non-conducting substances. They do not contain free electrons.

• When a dielectric is placed in an external electric field, its molecules align themselves in such a way that charges are induced on the surfaces of the dielectric with having charge density σp, and thus an electric field is created inside the dielectric.
• This induced electric field Ein has a value less than that of the external electric field. (Option (A), (B) and (C) are wrong.)

Both these fields have opposite directions.

Therefore the magnitude of the net electric field inside a dielectric is less than that of the external electric field.

The direction of the net electric field inside the dielectric is the same as that of the external electric field. Therefore option (D) is correct.

Given mass of shell m₁ = 200gm = 0.2Kg

Let its velocity after firing be v₁

mass of Gun m₂ = 4kg

its velocity after firing be v2

Momentum before firing = 0

Momentum after firing m₁v₁ + m₂v₂.

By Conservation of Momentum

m₁v₁ + m₂v₂ = 0

\(\Rightarrow v_{2}=\frac{m_{1} v_{1}}{m_{2}}\) .....(i)

The energy is conserved. So, the total Kinetic Energy of Gun and bullet is conserved and is equal to 1.04kJ.

\(\frac{1}{2} m_{1}\left(v_{1}\right)^{2}+\frac{1}{2} m_{2}\left(v_{2}\right)^{2}=1.05 \times 10^{3} \mathrm{~J}\) .....(ii)

Putting Eq (i) in Eq (ii) we get

\(\frac{1}{2} m_{1}\left(v_{1}\right)^{2}+\frac{1}{2} m_{2}\left(\frac{m_{1} v_{1}}{m_{2}}\right)^{2}=1050\)

Putting the values of m₁ and m₂ we get

\(\frac{1}{2} 0.2\left(v_{1}\right)^{2}+\frac{1}{2}\left(\frac{0.2 v_{1}}{2}\right)^{2}=1050\)

\(\Rightarrow \frac{v_{1}^{2}}{10}+\frac{v_{1}^{2}}{200}=1050\)

\(\Rightarrow \frac{20 v_{1}^{2}+v_{1}^{2}}{200}=1050\)

\(\Rightarrow v_{1}^{2}=\frac{1050 \times 200}{21}\)

⇒ v₁ = around 100 m/s

So, speed of shell is 100 m/s.

According to Lenz law, the polarity of induced EMF is such that it opposes the change in magnetic flux responsible for its production.

Magnetic flux depends on the area of the circular conductor. The magnetic flux (ϕ) enclosed by the circular loop can be given as ϕ = B × Area

If the coil is cut somewhere and then moved in and out of the magnetic field, EMF will be generated.

The basic cause of induced emf is the change in magnetic flux.

When a conductor is moved in and out of the magnetic field, magnetic flux across it changes. this causes the generation of EMF in the direction such that it opposes the change.