Physics Test 226

The correct answer is C.

Explanation:

• The liquid which has less relative density will have less density.
• Relative Density = Density of a Substance / Density of Water
• Less density of liquid means, more volume of the liquid needs to be displaced in order to balance the same weight.
• Therefore object will float with its maximum volume submerged under the liquid which has the least density.
• And object will float with its minimum volume submerged under the liquid which has the highest density.
• Therefore, object will float with its maximum volume submerged under the liquid C.

Thus, the object will float with its maximum volume submerged under the liquid C.

Concept:

An alternating current is an electric current that reverses direction periodically.

• The general equation for alternating emf is given by.
• V = V₀ sin (ωt + ϕ) --- (1)
• V = V₀ cos (ωt + ϕ) --- (2)
• RMS means Root-Mean-Square of instantaneous voltage 
• V₀ = Peak current, ω = frequency, t = time

Calculation: 

Given, V = 400 sin (100 πt)

Comparing given equation from equation 1,

Then, Peak current V0 = 400, frequency ω = 100π

Bragg’s Law:

Bragg's Law relates the angle θ (at which there is a maximum in diffracted intensity) to the wavelength of X-rays and the interlayer distance d between the planes of atoms/ions/molecules in the lattice.

Condition for constructive interference (maxima in the reflected).

nλ = 2d sinθ

λ = Wavelength of the X-ray

d = distance of crystal layers

θ = incident angle (the angle between the incident plane and the scattering plane)

n = integer (order)

This condition is known as Bragg’s law.

From the above equation,

Concept:

Bohr model:

The Lyman series: 

• It includes the lines emitted by transitions of the electron from an outer orbit of quantum number n2 > 1 to the 1st orbit of quantum number n₁ = 1.
• All the energy wavelengths in the Lyman series lie in the ultraviolet band.

The Balmer series:

• It includes the lines due to transitions from an outer orbit n₂ > 2 to the orbit n₁ = 2.
• Four of the Balmer lines lie in the "visible" part of the spectrum.

Paschen series (Bohr series, n₁ = 3)

Brackett series (n₁ = 4)

Pfund series (n₁ = 5)​

Paschen series: 

• When an electron in a Hydrogen atom transit from a higher energy orbit to 3rd orbit. (outer orbit n₂ = n > 3 to the orbit n₁ = 3) known as Paschen Series.
• So the empirical formula for the observed wavelengths (λ)for hydrogen (Z = 1) is

Calculation:

Given: 

The shortest wavelength in the Lyman series of the hydrogen spectrum = 912 Å.

CONCEPT:

• AND Gate: The Logic AND Gate is a digital logic circuit whose output HIGH only when all the inputs are 1 (HIGH) otherwise output will be LOW.

• OR Gate: The Logic OR Gate is a digital logic circuit whose output goes HIGH only when any one or more than one of its inputs is HIGH.

• NOT Gate: The Logic NOT Gate is the most basic of all the logic gates and is often referred to as an Inverting Buffer or simply an Inverter.

• NAND Gate: The logic Gate is obtained after adding NOT Gate after AND Gate.

• NOR Gate: The logic Gate is obtained after adding NOT Gate after OR Gate.

EXPLANATION:

For Circuit A: 

• The output of NAND gate 0 and the output of NOT is 1. 
• The output of the OR gate for input 1 and 0 is 1.

For Circuit B:

• The Output of NOT gate for 0 is 1 and of other NOT gate for input 1 is 0.
• The output of the NAND gate for input 1 and 0 is 1.

For Circuit C:

• The output for NOR Gate for input 1 and 1 is 0
• and the output of the AND gate for input 1 and 0 is 0.
• Hence the correct answer is option 2.

Explanation:

Young’s modulus

• The ratio of tensile (or compressive) stress (σ) to the longitudinal strain (ε) is defined as young’s modulus and is denoted by the symbol Y.

• Since strain is a dimensionless quantity, the unit of young’s modulus is the same as that of stress i.e., N m^–2 or Pascal (Pa).
• Metals have larger values of young’s modulus than alloys and elastomers.
• A material with large value of young’s modulus requires a large force to produce small changes in its length.
• The Young’s modulus and shear modulus are relevant only for solids since only solids have lengths and shapes.

Bulk modulus

• The ratio of hydraulic stress to the corresponding hydraulic strain is called bulk modulus.
• It is denoted by symbol B.

• The negative sign indicates the fact that with an increase in pressure, a decrease in volume occurs.
• That is, if p is positive, ∆V is negative.
• Bulk modulus is relevant for solids, liquid and gases.
• It refers to the change in volume when every part of the body is under uniform stress so that the shape of the body remains unchanged. 

CONCEPT:

For a sphere rolling with an angular velocity ω and linear velocity v,

The relation between them = v = Rω

The moment of inertia of a body is the quantity of the body which resists angular acceleration which is the sum of the product of the mass of every particle with its square of a distance from the axis of rotation.

Moment of inertia of a sphere I = (2/5)MR²

EXPLANATION:

let the velocity of the center is V₀ and if the radius is R and angular velocity ω₀ 

Then, V₀ = Rω 

The sphere is rolling without slipping.

Given,

V0 = 5 cm/s = 0.05 m/s

m = 500 g = 0.5 kg

K.E = (7/10) (0.5) (0.05)² = 8.75×10 ⁻⁴ J

Hence the correct answer is option 3.

Concept:

The formula for uniformly accelerated motion along a straight line (Position time relation) –

Where, s = displacement of the body, u = initial velocity, a = acceleration of the body and t = time

Explanation:

Given – Vertical distance (y) = 10 cm = 1/10 m, horizontal distance (x) = 10 cm = 1/10 m, number of steps (n) = 5 and g = 10 m/s²

∴ Distance covered in vertical direction is –

CONCEPT:

• The angular momentum of a particle rotating about an axis is defined as the moment of the linear momentum of the particle about that axis.
• It is measured as the product of linear momentum and the perpendicular distance of its line of action from the axis of rotation.
• It is analogous to linear momentum and its SI unit is kg m²/s.

• The relation between the angular momentum and moment of inertia is given by

L = Iω

Where I = moment of inertia, L = angular momentum, and ω = angular velocity.

• Moment of inertia of a circular ring about an axis passing through its center and perpendicular to its plane is

I = MR²

Where M = mass of the ring and R = radius of the ring

EXPLANATION:

Given – M₁ = M, M₂ = 2m, R₁ = R₂ = R (let), initial velocity (ω₁) = ω and final velocity (ω₂) = ω’

We have a thin circular ring of mass M and radius r and it is rotated at constant angular velocity ω

• The initial moment of inertia is of the circular ring is

• When another ring of the same mass is put on the 1st rotating ring, then the final moment of inertia is

• Since there is no external torque acting on this system, this means that angular momentum of the system will be conserved
• Thus by conservation of angular momentum

Concept:

Kepler's Law:

• Kepler’s laws of planetary motion are three scientific laws describing the motion of planets around the sun.

Kepler’s first law: 

• All the planets revolve around the sun in elliptical orbits having the sun at one of the foci.

Kepler’s second law states:

• The radius vector drawn from the sun to the planet sweeps out equal areas in equal intervals of time.
• Areal velocity is constant.
• Formula, Areal velocity, dA/dt = L/2m, where L = angular momentum, m = mass 

Kepler’s law of periods:

• The square of the time period of revolution of a planet around the sun in an elliptical orbit is directly proportional to the cube of its semi-major axis.
• Formula,  where r = radius, G = universal gravitational constant, M = mass

Calculation:

The radius of the geostationary satellite, r₁ = R + 7R = 8R

The time period on the earth, T₁ = 24h

The radius of the second satellite, r₂ = R + 3R = 4R

Here, R = radius of the earth

The relationship between the radius of the satellite and the time period is given as,