Mechanics & Motion — Part 4: 32 MCQs with Explanations

This is based on Newton's third law and conservation of momentum. When the person exerts a force on the ball to throw it forward, the ball exerts an equal and opposite force on the person, pushing them backward.

The time period of a simple pendulum (or swing) T = 2π√(L/g) depends only on its length (L) and the acceleration due to gravity (g). It is independent of the mass of the bob. Adding more mass does not change the time period.

Centripetal force F_c = mv²/r. For a constant mass (m) and radius (r), F_c ∝ v². If the speed (v) is doubled, the force becomes (2)² = 4 times the original force.

Boltzmann's constant (k) relates the average kinetic energy of particles in a gas with the temperature. Its value is 1.38 × 10⁻²³ joules per kelvin (J/K).

During a phase transition, the heat supplied (latent heat) is used to overcome the intermolecular forces to change the phase of the substance, not to increase the average kinetic energy of the molecules. Therefore, the temperature remains constant.

Amplitude is a measure of the energy carried by the wave and is independent of the wave's frequency, wavelength, or velocity. The other three are related by the equation v = fλ.

Momentum (p = mv) has units of kg m/s. A Newton-second (N·s) is equivalent to (kg m/s²) × s = kg m/s, so it is the same unit.

In Newton's second law of motion, F = ma, 'm' stands for the inertial mass of the object, which is a measure of its resistance to acceleration.

The value of 'g' is minimum at the equator. This is due to two reasons: the Earth's radius is larger at the equator (centrifugal force from rotation also reduces effective g), and the centrifugal force due to Earth's rotation is maximum at the equator.

X-rays are produced in an X-ray tube when high-speed electrons are accelerated and suddenly stopped by striking a metal target (usually tungsten or copper). The kinetic energy of the electrons is converted into X-ray radiation.

According to Newton's first law, if a body is moving with constant velocity (no acceleration), the net external force acting on it must be zero.

Inertia is the natural tendency of an object to resist changes in its state of motion. This property is quantified by its mass.

A force is non-conservative if the work done by it depends on the path taken. Friction is a classic example, as longer paths result in more negative work done. Work done by conservative forces (gravity, spring) is path-independent.

At the highest point, the vertical velocity of the ball becomes zero. Kinetic energy (KE = 1/2 mv²) is proportional to the square of velocity. Since v=0, the KE at that instant is zero.

Even with constant speed, the direction of velocity changes continuously. This change in direction constitutes an acceleration directed towards the center of the circle, called centripetal acceleration.

Mass is a fundamental property of matter and is a measure of its inertia. It is a scalar quantity that remains constant everywhere, regardless of location. Weight varies due to changes in 'g'.

The de Broglie hypothesis states that every moving particle has a wave-like nature. Its wavelength λ is given by λ = h/p, where h is Planck's constant and p is the momentum of the particle. Therefore, λ ∝ 1/p.

Impulse = Force × time = 10 N × 5 s = 50 Ns.

A photon is a massless particle. It always travels at the speed of light in vacuum and has zero rest mass. Its energy and momentum are carried by its frequency, not its mass.

According to Newton's law of gravitation, F ∝ 1/r². If r becomes 4r, then r² becomes 16r². Therefore, the new force F' ∝ 1/(16r²) = (1/16) × (1/r²). So, the force becomes one-sixteenth of the original.

The centripetal force is always perpendicular to the instantaneous displacement of the body. Work done W = F·d = F d cos 90° = 0. This is true for any displacement along the circular path.

The range R = (u² sin 2θ)/g. Since sin(2θ) = sin(180° - 2θ), the range is the same for complementary angles (θ and 90° - θ). 30° and 60° are complementary (30+60=90).

Escape velocity v_esc = √(2GM/R). It depends on the mass (M) and radius (R) of the planet, and the gravitational constant (G). It is independent of the mass of the body trying to escape.

At maximum height, final velocity v=0. Using v² = u² - 2gh, we get 0 = (20)² - 2×10×h => 400 = 20h => h = 20 m.

Stopping distance is proportional to the square of the initial speed (from v² = 2as). If speed is halved (v/2), the distance becomes (1/2)² = 1/4 times the original distance.

Orbital velocity v₀ = √(GM/R). Since g = GM/R², we have GM = gR². Substituting, v₀ = √(gR²/R) = √(gR). Escape velocity is √(2gR).

Total energy E = (1/2)kA². Potential energy U = (1/2)kx². For U = E/2, we have (1/2)kx² = (1/2) × (1/2)kA² => x² = A²/2 => x = A/√2.

Work done against gravity = mgh = 10 kg × 10 m/s² × 5 m = 500 J.

Let initial p = p. New p' = 1.5p. Initial K = p²/2m. New K' = (1.5p)²/2m = 2.25p²/2m = 2.25K. % increase = (2.25-1)×100 = 125%.

By conservation of momentum, m_b v_b = m_g v_g => v_g = (m_b/m_g)v_b. KE_b / KE_g = (1/2 m_b v_b²) / (1/2 m_g v_g²) = (m_b/m_g) × (v_b/v_g)² = (m_b/m_g) × (m_g/m_b)² = m_g/m_b. So, KE_b: KE_g = m_g: m_b.

At the highest point for critical speed (minimum speed to complete the circle), the centripetal force is provided entirely by the weight of the body. The tension (T) in the string becomes zero. mv²/r = mg.

A conservative force has the property that the work done by it around any closed path is zero. Equivalently, the work done is independent of the path and depends only on the initial and final positions.