Physics Test 232

Concept:

The moment of inertia of a solid object describes how difficult it is to change its rotational motion about a given axis.

For a spherical shell with inner radius  R₁ and outer radius R₂, the moment of inertia can be calculated by considering the distribution of mass in this shell.

The moment of inertia of a thin spherical shell about its center of mass is given by,

Concept:

Refraction through a Prism:

• When light passes through a prism, it bends due to a change in speed as it moves from one medium to another.
• The angle of deviation (δ) is the angle between the incident ray and the emergent ray.
• For a prism, the relationship between the angle of incidence (i), angle of prism (A), and angle of deviation (δ) is given by the prism formula:
• δ = i + e − A
• At minimum deviation, the angle of incidence equals the angle of emergence (i = e).
• δ_m = 2i − A
• The refractive index (n) of the prism is related to the angle of prism (A) and the angle of minimum deviation (δm) by:

• Refractive Index (n):
  • Definition: A measure of how much the speed of light is reduced inside a medium.
  • SI Unit: Dimensionless
  • Dimensional Formula: None

Concept:

In steady-state heat conduction through a composite slab, the rate of heat transfer is the same through each layer of the slab.

The total heat transfer rate can be calculated by considering the thermal resistances of the two layers.

The heat transfer through each material follows Fourier’s law of heat conduction,

Q = ΔT / R_total

Where:

• Q is the rate of heat transfer,
• ΔT is the temperature difference across the slab,
• R_total is the total thermal resistance of the composite slab.

The thermal resistance R of a slab of thickness d and thermal conductivity K is given by:

R = d / (K ⋅ A)

Calculation:

The thermal resistance of the first layer (with thickness x and thermal conductivity K) is:

⇒ R₁ = x / (K ⋅ A)

The thermal resistance of the second layer (with thickness 4x and thermal conductivity 2K) is:

⇒ R₂ = 4x / (2K ⋅ A) = 2x / (K ⋅ A)

The total thermal resistance of the composite slab is:

⇒ R_total = R₁ + R₂ = x / (K ⋅ A) + 2x / (K ⋅ A) = 3x / (K ⋅ A)

The rate of heat transfer through the slab is:

⇒ Q = ΔT / R_total = (T₂ - T₁) / (3x / (K ⋅ A))

⇒ Q = (K ⋅ A ⋅ (T₂ - T₁)) / (3x)

The heat transfer rate is given as:

⇒ Q = A(T₂ - T₁)Kx ⋅ f

Comparing this with the expression for Q, we get:

f = 1/3

∴ The correct option is 4

Explanation:

Instantaneous Speeds

• The speed in SHM is given by,
• v(t) = ω √(A² - x²(t))
• Since the displacement is the same at both t₁ and t₂, and SHM is symmetric about the equilibrium position, the magnitudes of the speed at both times will be the same, but the directions will be opposite.
• Hence, the instantaneous speeds (in terms of magnitude) will be equal.

Instantaneous Accelerations

• The acceleration in SHM is given by,
• a(t) = -ω² x(t)
• Since the displacement x(t₁) = x(t₂), the acceleration at t₁ and t₂ will be the same, both in magnitude and direction.
• Hence, the instantaneous accelerations will be equal.

Kinetic Energies

• The kinetic energy in SHM is given by,
• KE = (1/2) m v²
• Since the speeds (in magnitude) are the same at t₁ and t₂, the kinetic energy, which depends on the square of the speed, will also be the same at both instants.
• Hence, the kinetic energies will be equal.

Therefore, all of these (speeds, accelerations, and kinetic energies) will be equal at the two closest instants when the displacement is the same.

The correct answer is: 4) All of these

Concept:

When a charged particle moves through a magnetic field, it experiences a magnetic force perpendicular to both its velocity and the magnetic field.

This force acts as a centripetal force, causing the particle to move in a circular path. The magnetic force is given by:

F_B = qvB

Where:

• q is the charge of the particle.
• v is the velocity of the particle.
• B is the magnetic field strength.

The centripetal force required for circular motion is given by:

Where:

• m is the mass of the particle.
• r is the radius of the circular path.

For the particle to just pass through the magnetic field and reach ( x > b ), the radius of the circular path must be equal to or greater than the width of the magnetic field region (i.e r ≥ (b − a).

Concept:

• The maximum height H of a projectile launched with an initial speed v0 and launch angle θ is given by:
• H = (v₀² ⋅ sin² θ) / (2g)

where g is the acceleration due to gravity.

• The Range of a Projectile(R) is given by:
• R = (v₀² ⋅ sin ²θ) / g

Calculation:

Since sin ²θ = 2 sin θ cos θ, we have:

⇒v₀² = Rg / (2 sin θ cos θ)

Substitute into Height Formulas

For height h at angle θ:

⇒h = (v₀² ⋅ sin² θ) / (2g)

Substituting v₀²:

⇒h = (R sin θ) / (4 cos θ)      ________(1)

For height h' at angle 90° - θ:

⇒h' = (v₀² ⋅ cos2 θ) / (2g)

Substituting v₀²:

⇒h' = (R cos θ) / (4 sin θ)  __________(2)

Relationship Between Heights from (1) and (2)

⇒h h' = (R sin θ / (4 cos θ)) ⋅ (R cos θ / (4 sin θ))

Simplifying gives:

⇒h h' = (R²) / 16

∴ The correct option is 1.

Concept:

When a nucleus disintegrates into two nuclear parts, the principle of conservation of momentum is applied.

According to this law, the total momentum before and after disintegration remains the same.

The momentum of each part is given by,

Momentum = mass × velocity

Calculation:

Let the masses of the two nuclear parts be ( m₁) and ( m₂), and their velocities be ( v₁) and ( v₂) respectively.

According to the conservation of momentum,

m₁v₁ = m₂v₂

From the problem, the velocities of the two parts are in the ratio ( 2:1)

Concept:

Since power is constant and P = F · v, we can express the force F as: F = P / v

According to Newton's Second Law: F = m · a

⇒m · a = P / v

⇒a = P / (m · v)

Acceleration a is also the rate of change of velocity with respect to distance s: a = v dv/ds

⇒v dv/ds = P / (m · v)

⇒ v² dv/ds = P / m

Integrate both sides with respect to s

⇒∫ v² dv = (P / m) ∫ ds

The integral on the left side is

⇒ v³/3 = (P / m) s + C

To determine the constant C, use the initial condition.

⇒At s = 0, the velocity is v1: v13/3 = C

Thus, the equation becomes: v³/3 = (P / m) s + v₁³/3

v³ = 3(P / m) s + v₁³

Finally, take the cube root to solve for the final velocity v₂: v₂ = [3(P / m) s + v₁³]^1/3

The final velocity v₂ of the car after traveling a distance s under a constant power P is given by: v₂ = [3(P / m) s + v13]^1/3

∴ The correct option is 4

Concept:

The position vector of the particle is given as: r(t) = r₀(1 - at)t,

Where:

• r₀ is a constant vector
• a is a constant
•  The particle returns to its initial position when r(t) = 0.

Calculation:

At t = 0, the initial position is:

⇒ r(0) = r₀(1 - a(0))0 = 0

Time when the particle returns to the starting point

⇒ r(t) = 0, we have,

⇒ r₀(1 - at)t = 0

⇒ t = 0 (initial point) or t = 1/a

Thus, the particle returns to the starting point at

⇒ t = 1/a.

The velocity vector is given by:

⇒ v(t) = d(r(t))/dt = r₀(1 - 2at)

⇒ |v(t)| = |r₀| |1 - 2at|

The total distance covered is the integral of the velocity magnitude over time:

Distance = ∫₀^1/a |r₀| |1 - 2at| dt

For t ∈ [0, 1/2a], 1 - 2at is positive
For t ∈ [1/2a, 1/a], 1 - 2at is negative
 

Distance = |r₀| ( ∫₀^1/2a (1 - 2at) dt + ∫_1/2a^1/a -(1 - 2at) dt )

⇒ ∫₀^1/2a (1 - 2at) dt = 1/4a

⇒ ∫_1/2a^1/a -(1 - 2at) dt = 1/4a

⇒ |r₀| (1/4a + 1/4a) = |r₀|/2a

∴ The correct option is 3

Concept:

Motion of Each Particle

The position x(t) of a particle in SHM is described by:

x(t) = A cos(ω t + φ)

where ω is the angular frequency and φ is the phase constant.

The angular frequency ω is related to the period T by ω = 2π / T

Calculation:

Here,

Particle P: Period T_P = 3 s

Particle Q: Period T_Q = 6 s

Angular Frequencies

⇒ for particle P: ω_P = 2π / T_P = 2π / 3 rad/s

⇒For particle Q: ω_Q = 2π / T_Q = π / 3 rad/s

The speed v of a particle in SHM is given by:

⇒v = ω √(A² - x²)

At the meeting point, the speeds of particles P and Q are:

⇒v_P = ω_P √(A² - x²) v_Q = ω_Q √(A² - x²)

The ratio of the speeds is:

⇒v_P / v_Q = ω_P / ω_Q

Substitute the angular frequencies:

⇒v_P / v_Q = (2π / 3) / (π / 3) = 2

The ratio of the speeds of particles P and Q when they meet is 2.

∴ The correct option is 2