Mechanics & Motion — Part 3: 40 MCQs with Explanations

Δx Δp ≥ ħ/2, where ħ = h/2π. Some texts use Δx Δp ≥ h/4π which is equivalent.

Derived quantities are those that are expressed in terms of base quantities. Velocity is derived from length and time (m/s), whereas length, time, and mass are base quantities in the SI system.

Scalar quantities are described by magnitude only. Speed is the magnitude of velocity and has no direction. Force, velocity, and displacement are vector quantities.

The magnitude of the resultant R = √(F² + F² + 2F² cos θ). For θ=120°, cos 120° = -1/2. So, R = √(2F² + 2F²(-1/2)) = √(2F² - F²) = √(F²) = F.

Uniform motion implies constant velocity, which means both constant speed and constant direction. This is best described as equal displacements in equal intervals of time, regardless of how small the time interval is.

Using the second equation of motion for constant acceleration: s = ut + (1/2)at². Here, u=0, a=2 m/s², t=4 s. So, s = 0 + (1/2)×2×(4)² = (1/2)×2×16 = 16 m.

Acceleration is defined as the rate of change of velocity. On a velocity-time graph, the slope (Δv/Δt) directly gives the acceleration of the object.

At the highest point, the ball momentarily comes to rest, so its velocity is zero. However, the acceleration due to gravity (g) acts constantly downwards throughout its motion, so at the highest point, a = g downwards.

In the absence of air resistance, all objects fall with the same constant acceleration due to gravity (g), regardless of their mass. Therefore, the ratio of their accelerations is 1:1.

Impulse (J) = Force × time = Change in momentum (Δp). Therefore, Δp = F × t = 10 N × 3 s = 30 Ns.

Newton's first law states that an object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. This property of objects to resist changes in their motion is called inertia.

Centripetal force acts towards the center of the circle, which is perpendicular to the direction of the instantaneous displacement (tangential) of the body. Work done W = F·d = F d cos θ. Since θ = 90°, cos 90° = 0, so work done is zero.

Kinetic energy K = p²/2m. If K becomes 4K, then p²/2m = 4 × (p_initial²/2m). This implies p² = 4 p_initial², so p = 2 p_initial. Momentum becomes double.

Potential energy P.E. = mgh. Given P.E. = 1 J, m = 1 kg, and g ≈ 10 m/s². Then h = P.E. / (mg) = 1 / (1×10) = 0.1 m.

Angular momentum (L) is conserved when the net external torque acting on the system is zero (τ = dL/dt = 0). Force can act without producing torque if applied at the axis or radially.

Even with constant angular speed, the direction of the linear velocity vector changes continuously. This change in direction requires a centripetal acceleration (v²/r) directed towards the center. Tangential acceleration is zero only if angular speed is constant.

Moment of inertia (I) is the rotational analog of mass. It depends on the mass of the body and how that mass is distributed relative to the axis of rotation (I = Σ mr²).

Linear momentum p = mass × velocity = 2 kg × 3 m/s = 6 kg m/s.

For a body moving in a circle of radius 'r', the linear velocity (v) is related to the angular velocity (ω) by the equation v = rω.

The acceleration due to gravity (g) inside the Earth is given by g' = g (1 - d/R), where d is the depth. At the center, d = R, so g' = g (1 - 1) = 0.

The orbital velocity for a satellite close to Earth is given by v = √(GM/R). For Earth, this value is approximately 7.9 km/s, which is rounded to 8 km/s. The escape velocity is 11 km/s.

When the elevator accelerates upwards, the normal reaction from the floor (which is the apparent weight) increases. This is because the floor must push up with extra force to accelerate the person upwards. Apparent weight = m(g + a).

In SHM, velocity v = ω√(A² - x²). The velocity is maximum when the displacement x = 0 (at the mean position). So, v_max = ω√(A² - 0) = ωA.

In SHM, acceleration is directly proportional to the negative of displacement. a = -ω²x. The magnitude of acceleration is proportional to the displacement from the mean position.

In vacuum, all electromagnetic waves (radio, light, X-rays, etc.) travel at the same constant speed, denoted by 'c', which is approximately 3.00 × 10⁸ m/s.

Resistance R = ρL/A. It depends on the material's resistivity (ρ), length (L), and cross-sectional area (A). Temperature affects resistivity. Resistance does not directly depend on the mass of the wire.

The magnetic force is given by F = qvB sin θ. The value of sin θ is maximum (1) when θ = 90°. So, the force is maximum when the charge moves perpendicular to the magnetic field.

Fleming's left-hand rule (for motors) gives the direction of force acting on a current-carrying conductor placed in a magnetic field. The thumb, forefinger, and middle finger are mutually perpendicular, representing Force, Field, and Current.

Lenz's law states that the induced current opposes the change in magnetic flux that produced it. This opposition means that work must be done against this opposing force, and this work is converted into electrical energy, thus conserving energy.

Kinetic energy K = (1/2)mv². K is proportional to v². If v becomes 2v, then K becomes (2)² = 4 times the original kinetic energy.

The average translational kinetic energy of gas molecules is directly proportional to the absolute temperature. KE_avg = (3/2)kT, where k is Boltzmann's constant.

The first law of thermodynamics (ΔU = Q - W) is essentially the law of conservation of energy applied to thermodynamic systems. It states that the change in internal energy of a system equals the heat added to the system minus the work done by the system.

Drift velocity is the average velocity attained by charged particles in a material due to an electric field. It is very slow, typically on the order of millimeters per second (10⁻⁴ to 10⁻³ m/s).

The magnetic force is F = qvB sin θ. For perpendicular motion, θ = 90°, so sin 90° = 1. Therefore, F = 5 C × 2 m/s × 0.1 T × 1 = 1 N.

According to Newton's second law, F = ma. Therefore, a = F/m = 20 N / 5 kg = 4 m/s².

The range of a projectile is given by R = (u² sin 2θ)/g. The value of sin 2θ is maximum (1) when 2θ = 90°, so θ = 45°. At this angle, the range is maximum.

A body thrown horizontally from a height has an initial horizontal velocity but its vertical motion is under the influence of gravity (constant acceleration). This combination of uniform horizontal motion and accelerated vertical motion results in a parabolic trajectory.

Escape velocity (v_esc) is the minimum speed needed for an object to escape from a planet's gravitational influence without further propulsion. For Earth, v_esc = √(2GM/R) ≈ 11.2 km/s.

Since K = p²/2m, we have p = √(2mK). If K is constant, p ∝ √m. Therefore, p1/p2 = √(m1/m2) = √(2/4) = √(1/2) = 1/√2. So the ratio is 1: √2.

According to Coulomb's law, F ∝ 1/r². If r becomes r/2, then r² becomes (r/2)² = r²/4. Therefore, the new force F' ∝ 1/(r²/4) = 4 × (1/r²). So, the force becomes four times.