Physics Test - 8

Equilibrium of a rigid body:

A rigid body is said to be in mechanical equilibrium if both its linear momentum and angular momentum are not changing with time, or equivalently, the body has neither linear acceleration nor angular acceleration.

Condition for the mechanical equilibrium:

• The total force, i.e. the vector sum of the forces, on the rigid body is zero.
• The total torque, i.e. the vector sum of the torques on the rigid body is zero.

If the forces on a rigid body are acting in the 3 dimensions, then six independent conditions to be satisfied for the mechanical equilibrium of a rigid body.

If all the forces acting on the body are coplanar, then we need only three conditions to be satisfied for mechanical equilibrium.

A body may be in partial equilibrium, i.e., it may be in translational equilibrium and not in rotational equilibrium, or it may be in rotational equilibrium and not in translational equilibrium.

According to Einstein's mass energy equivalence:

\(E=m c^{2}\)

\(=10^{-6} \times 10^{-3} \times\left(3 \times 10^{8}\right)^{2} J\)

\(=9 \times 10^{7} J\)

As we know, 1 kilowatt hour \((1 {kWh})=3.6 \times 10^{6} {~J}\)

\(\therefore E=9 \times 10^{7} J \times \frac{1 k W h}{3.6 \times 10^{6} J}\)

\(=\frac{900}{36} k W h\)

\(=25 k W h\)

The rigidity modulus of a diamond is greater than any known matter. The modulus of rigidity, also known as shear modulus, is defined as a material property with a value equal to the shear stress divided by the shear strain. The general formula of shear modulus is written as shown:

Where τ is the shear stress in a given matter, which has the unit of force divided by area (N/m²), γ is the shear strain which does not have a unit (strain is the change of length divided by the original length), and G is the shear modulus or the modulus of rigidity, which has the unit of force divided by the area.

Newton's law of gravitation is applicable in the case of all bodies in the universe.

Newton's law of universal gravitation states that any two bodies in the universe attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Since any two bodies can be there so it doesn't matter whether they are small or large.

Diffraction of light cannot convert unpolarized light into polarized light.

• Polarization is defined as a phenomenon caused due to the wave nature of electromagnetic radiation.
• Sunlight travels through the vacuum to reach the Earth, which is an example of an electromagnetic wave.
• These waves are called electromagnetic waves because they form when an electric field that interacts with a magnetic field.
• There are two types of waves that are transverse waves, and longitudinal waves.

The intensity of EM waves is basically the number of photons emanating from the source. The speed of an EM wave in a dielectric medium is a function of the permittivity and permeability of the medium. Therefore, irrespective of the energy of a photon, if it has to pass through the medium, its propagation speed being a function of the medium rather than the source, remains constant.

Given,

Diameter of the sphere \(=2.4m\)

\(\therefore\) Radius of sphere, \(r=\frac{2.4}{2}=1.2 m\)

Surface charge density of conducting sphere,

\(\sigma=80 \times 10^{-6} C / m ^{2}\)

Therefore,

Charge on sphere will be:

\(q=\sigma A=\sigma 4 \pi r^{2}\)

\(q=80 \times 10^{-6} \times 4 \times 3.14 \times(1.2)^{2}\)

\(q=1.45 \times 10^{-3} C\)

Then, the total electric flux leaving the surface of the sphere will be calculated using the gauss formula, i.e.,

\(\phi=\frac{q}{\varepsilon_{0}}\)

\(\phi=\frac{1.45 \times 10^{-3}}{8.854 \times 10^{-12}} \quad\left(\because \epsilon_{0}=8.854 \times 10^{-12}\right)\)

\(\phi=1.6 \times 10^{8} Nm { }^{2} / C\)

Let \(C\) be the capacitance of the infinite ladder.

As the ladder is infinite, the addition of one more element of two capacitors \((1 \mu F\) and \(2 \mu F)\) across the points \(X\) and \(Y\) should not change the total capacitance.

Therefore, the total capacity of the arrangement shown in the figure must remain \(C\) only.

In figure, \(2 \mu F\) capacitor is in series with capacitance \(C\).

\(\therefore\) Their combined capacity \(=\frac{2 \times C}{2+C}\)

This combination is in parallel with \(1 \mu F\) capacitor.

The equivalent capacity of the arrangement is

\(1+\frac{2 C}{2+C}=C \)

\(\text { or, } C^{2}+2 C=2+3 C \)

\(\text { or, } C^{2}-C-2=0 \)

\(\therefore C=2,-1\)

As capacitance cannot be negative.

\(\therefore C=2 \mu F\)

Concept:

Force = T - W

ma = T - W

Therefore, \(a=\frac{T-W}{m}\)

Calculation:

Given:

W = \(1000\) kg, v = \(4\) m/s, t = \(2\) sec

Acceleration is given as

\(a=\frac{4-0}{2}=2 \mathrm{~m} / \mathrm{s}^{2}\)

Tension is given as

\(\mathrm{T}=\mathrm{mg}+\mathrm{ma}=\mathrm{m}(\mathrm{g}+\mathrm{a})\)

\(\mathrm{T}=1000(10+2)\)

\(\mathrm{T}=12000 \mathrm{~N}\)

An adiabatic expansion has less work done and no heat flow, thereby a lower internal energy comparing to an isothermal expansion which has both heat flow and work done.For an adiabatically expanding ideal monatomic gas which does work on its environment (W is positive), internal energy of the gas should decrease.