<p><em>Parhai — Free Punjab Board Notes</em></p>
<h2>Key Concepts</h2>
<h3>1. What Is a Wave?</h3>
<p>A wave is a mechanism that carries energy from one place to another through a disturbance in the medium — without the medium itself traveling along. Some waves are visible (water ripples), while others are invisible and detected only with sensitive instruments. Waves matter in daily life because:</p>
<ul>
<li>Sound reaches our ears in the form of waves.</li>
<li>Sunlight and heat travel to us as waves.</li>
<li>Radio and television broadcasting works through waves.</li>
<li>Medical imaging — broken bones, tumors, bullets inside the body — is detected using waves.</li>
</ul>
<h3>2. Wave Motion Explained</h3>
<p>Wave motion happens when a disturbance makes the particles of a medium vibrate to and fro about their mean positions at regular time intervals, and that disturbance is passed from one particle to the next until it reaches the far end of the medium.</p>
<p><strong>Water tub experiment:</strong> Dip the tip of a pencil into a tub of water and move it rapidly up and down. Ripples spread outward. If you float small pieces of paper on the surface, each piece only bobs up and down about its own position — it never moves forward with the wave. The disturbance travels; the water particles stay behind, vibrating in place.</p>
<p><strong>String experiment:</strong> Tie one end of a string to a hook and mark it with colored spots at equal intervals. Shake the free end up and down. Waves run down the string, while the colored marks simply vibrate about their mean positions instead of traveling with the wave.</p>
<h3>3. Simple Harmonic Motion (SHM)</h3>
<p>When a body vibrates in such a way that its <strong>acceleration is directly proportional to its displacement from the mean position</strong> and is <strong>always directed toward the mean position</strong>, the motion is called simple harmonic motion.</p>
<p><strong>Mass attached to a spring (proof):</strong> Imagine a mass <em>m</em> attached to a spring on a smooth horizontal surface, with the other end fixed to a wall. With no stretch, the mass rests at the equilibrium position O. Pull it to position A (an extreme), stretching the spring by a distance <em>x</em>.</p>
<ul>
<li><strong>Hooke's law:</strong> The applied force is directly proportional to the stretch: F = kx, where <em>k</em> is the spring constant.</li>
<li><strong>Spring constant:</strong> k = F/x — force per unit extension. Its unit is N/m.</li>
<li><strong>Restoring force:</strong> When released, the spring pulls the mass back toward O with force F = −kx (the negative sign shows it points opposite to the displacement, toward the mean position).</li>
</ul>
<p>By Newton's second law, F = ma. Equating the two expressions for force: ma = −kx, so a = −(k/m)x. Since k/m is constant, acceleration is proportional to displacement <em>x</em>, and the negative sign keeps it aimed at the mean position — exactly the definition of SHM.</p>
<p>As the mass returns toward O, displacement shrinks, so acceleration shrinks too. At O, x = 0, acceleration is zero, and velocity is maximum. Inertia carries the mass past O to the other extreme A′, compressing the spring; acceleration again points toward O and slows the mass until it momentarily stops at A′. The restoring force then pulls it back, and the cycle repeats between A and A′.</p>
<p><strong>Time period of mass-spring system:</strong> T = 2π√(m/k)</p>
<h3>4. The Simple Pendulum</h3>
<p>A simple pendulum is a small bob of mass <em>m</em> hung from a frictionless support by a light, inextensible string of length <em>l</em>.</p>
<p><strong>Why its motion is SHM:</strong> Pull the bob from its rest position O to point A and release it. Gravity pulls it back toward O. At O its velocity is maximum, so inertia carries it on to point B on the other side, where it stops momentarily. The whole way, whether moving from A to O (speeding up) or O to B (slowing down), the acceleration points toward O and grows or shrinks with the displacement — so the pendulum executes SHM between A and B.</p>
<p><strong>Energy changes:</strong> At the mean position O (lowest point), potential energy is minimum and kinetic energy is maximum. At the extremes A and B (highest points), potential energy is maximum and kinetic energy is zero. Between them, energy is part potential and part kinetic, but the total stays constant.</p>
<p><strong>Time period:</strong> T = 2π√(l/g), where <em>l</em> is the distance from the point of suspension to the center of the bob. It depends on the length and on <em>g</em>, but <strong>not on the mass of the bob</strong>.</p>
<h3>5. Hallmarks of SHM</h3>
<ul>
<li>The body vibrates to and fro about its mean position.</li>
<li>Acceleration always points toward the mean position and is proportional to displacement.</li>
<li>Acceleration is zero at the mean position and maximum at the extremes.</li>
<li>Velocity is maximum at the mean position and zero at the extremes.</li>
</ul>
<h3>6. Essential Terms</h3>
<ul>
<li><strong>Amplitude:</strong> The maximum displacement from the mean position to either extreme.</li>
<li><strong>Vibration:</strong> One complete round trip of the vibrating body about its mean position.</li>
<li><strong>Time period (T):</strong> The time needed to complete one vibration.</li>
<li><strong>Displacement:</strong> The distance of the vibrating body from the mean position at any instant.</li>
<li><strong>Periodic motion:</strong> Any motion that repeats itself after a regular time interval.</li>
</ul>
<h3>7. Mechanical vs Electromagnetic Waves</h3>
<p><strong>Mechanical waves</strong> need a material medium to be produced and to travel — for example, water waves and waves on a stretched string. <strong>Electromagnetic waves</strong> need no medium at all — for example, radio waves, heat and light waves, and X-rays.</p>
<h3>8. Two Types of Mechanical Waves</h3>
<p><strong>Transverse waves:</strong> The particles of the medium vibrate <em>perpendicular</em> to the direction in which the wave travels. Examples: ripples on water (particles move up and down while the wave moves outward), and waves on a shaken string.</p>
<p><strong>Longitudinal (compressional) waves:</strong> The particles vibrate <em>parallel</em> to the direction of wave travel. Examples: sound waves, and the waves produced in a mass attached to a spring.</p>
<h3>9. Anatomy of a Wave</h3>
<ul>
<li><strong>Crest:</strong> The part of a transverse wave where particles rise above the normal position.</li>
<li><strong>Trough:</strong> The part where particles dip below the normal position.</li>
<li><strong>Wavelength (λ):</strong> The distance between two consecutive crests (or two consecutive troughs). One crest plus one trough makes one complete wave.</li>
<li><strong>Frequency (f):</strong> The number of waves passing a point in one second. Its unit is the hertz (Hz), also written as cycles/sec or vibrations/sec.</li>
</ul>
<h3>10. Waves as Carriers of Energy</h3>
<p>Energy moves from place to place in two ways: <strong>through matter</strong> (a fired bullet carries kinetic energy to its target; dam water carries kinetic energy to turbine blades that generate electricity) and <strong>through waves, without any matter being transported</strong>.</p>
<p>Try this: float a cork on water in a tray and produce waves with a moving rod at the far end. The waves reach the cork and make it bob up and down — the energy you spent moving the rod has traveled to the cork, yet the water particles themselves never moved forward. They only vibrated in place as the wave passed through.</p>
<h3>11. Four Properties of Waves</h3>
<p>Waves show four characteristic behaviors: <strong>reflection, refraction, interference, and diffraction</strong>. All four can be demonstrated with the ripple tank.</p>
<h3>12. The Ripple Tank</h3>
<p>A ripple tank is the standard apparatus for producing water waves and studying their properties. It is a rectangular tray with a glass bottom, raised about half a metre above the table on four legs. A vibrator (an oscillating electric motor on a wooden bar) drives a plate whose lower edge just touches the water; rubber bands suspend the plate so it can move freely.</p>
<p><strong>Straight waves:</strong> When the vibrator runs, the plate produces straight waves. A lamp above the tray projects their image onto white paper below. Crests act like convex lenses — they converge light and appear as bright lines. Troughs act like concave lenses — they diverge light and appear as dark bands between the bright lines.</p>
<p><strong>Circular waves:</strong> Raise the bar and lower a small knob until it just touches the water. Switching on the vibrator now produces expanding circular waves.</p>
<h3>13. The Stroboscope — Freezing Waves</h3>
<p>A stroboscope is a device that makes moving waves appear stationary. It is a rotating disc with slits around its edge. You look through the slits at the wave: each slit lets you glimpse the wave only at one instant. If the disc speed is adjusted so that each new slit reaches your eye exactly when the wave has advanced by one wavelength, every glimpse shows the wave in the same position — the wave looks frozen.</p>
<h3>14. The Wave Equation: v = fλ</h3>
<p>Watch a floating piece of paper: it vibrates up and down but does not travel. In one vibration (time T), successive crests pass a point while the wave covers one wavelength λ. If the wave speed is v, the distance covered in time T is vT, which equals λ:</p>
<p>λ = vT, so v = λ/T. Since 1/T = f, we get <strong>v = fλ</strong> — wave speed equals frequency times wavelength. The vibrator sets the frequency; the speed and wavelength are decided by the properties of the medium.</p>
<h3>15. Reflection of Waves</h3>
<p>Water waves reflect according to the same laws as light. Generate straight waves in the ripple tank and place a flat obstacle at an angle to their path. The waves bounce off in a definite direction. Measuring with a stroboscope and protractor, the <strong>angle of incidence (∠i)</strong> between the incoming ray and the normal equals the <strong>angle of reflection (∠r)</strong> between the outgoing ray and the normal.</p>
<h3>16. Refraction of Waves</h3>
<p>Refraction is the bending of waves from their original path when they enter a different medium. Water-wave speed depends on depth: waves slow down in shallow water. Cover half the ripple tank's bottom with a thick glass plate so one half is shallow. Waves entering the shallow part keep the same frequency (fixed by the vibrator) but their wavelength shrinks, since v = fλ and v has decreased. If the boundary between the deep and shallow parts meets the waves at an angle, the waves also change direction — bending toward the normal when entering the shallower region. That change of path is refraction.</p>
<h3>17. Diffraction</h3>
<p>Diffraction is the bending of waves around corners or obstacles. Place two obstacles in a line in the ripple tank with a gap between them about equal to the wavelength. Waves passing through the gap spread out in all directions and turn circular. Diffraction is clear only when the slit (or obstacle) is roughly the same size as the wavelength; a much wider slit lets the waves continue straight with only slight bending at the edges.</p>
<h3>18. Interference</h3>
<p>When the particles of a medium are acted upon by two waves at the same time, the combined effect is called interference.</p>
<ul>
<li><strong>Constructive interference:</strong> Crest meets crest (or trough meets trough). The two waves reinforce each other, and the resulting amplitude doubles.</li>
<li><strong>Destructive interference:</strong> Crest meets trough. The waves cancel each other, leaving the medium's surface undisturbed.</li>
</ul>
<p><strong>Ripple tank demo:</strong> Attach two knobs to the vibrating bar so they both touch the water. Two sets of identical circular waves (same frequency, same amplitude, started together) spread out and overlap — interference patterns become visible where they meet.</p>
<h3>19. Stationary Waves, Nodes and Antinodes</h3>
<p>When two identical waves travel along the same line in opposite directions, they produce <strong>stationary waves</strong> at certain frequencies. Fasten one end of a string to a support and shake the free end up and down; increasing the shaking rate makes the string vibrate with maximum amplitude at particular frequencies — these are stationary waves (the incident wave interferes with the wave reflected from the support).</p>
<p><strong>Conditions:</strong> the two waves must be identical in every respect and must travel in opposite directions along the same line.</p>
<ul>
<li><strong>Nodes:</strong> points where the amplitude is always zero (N₁, N₂, N₃...).</li>
<li><strong>Antinodes:</strong> points where the amplitude is maximum (A₁, A₂, A₃...).</li>
</ul>
<p>The wavelength of a stationary wave is <strong>twice the distance between two consecutive nodes</strong> (or two consecutive antinodes).</p>
<h3>20. Fundamental Frequency and Harmonics</h3>
<p>At the lowest frequency the string forms a single loop — this is the <strong>fundamental frequency</strong> (first harmonic). With two loops the frequency is the second harmonic (first overtone), equal to twice the fundamental: f₂ = 2f₁. In general, the nth harmonic is <strong>f_n = nf₁</strong>.</p>
<h2>Questions & Answers</h2>
<h3>Q1. Define vibratory motion and state its conditions.</h3>
<p>The to-and-fro motion of a body about its mean position — with the particles of the medium staying in place — is vibratory motion. Its conditions: (1) a restoring force acts on the body; (2) the motion repeats after regular intervals; (3) it obeys the law of conservation of energy; (4) the body moves to and fro about its mean position.</p>
<h3>Q2. State the characteristics of SHM.</h3>
<p>(1) The body always vibrates about its mean position. (2) Acceleration is always directed toward the mean position. (3) Acceleration is proportional to displacement — zero at the mean position, maximum at the extremes. (4) Velocity is maximum at the mean position and zero at the extremes.</p>
<h3>Q3. What is Hooke's law? Define the spring constant.</h3>
<p>Hooke's law: the force applied to a spring is directly proportional to the increase in its length (F ∝ x). The spring constant k = F/x is the ratio of applied force to extension; its unit is N/m.</p>
<h3>Q4. Distinguish transverse and longitudinal waves with examples.</h3>
<p>In transverse waves, medium particles vibrate perpendicular to the wave's direction (water ripples, string waves). In longitudinal (compressional) waves, particles vibrate parallel to the direction of travel (sound waves, spring-mass waves).</p>
<h3>Q5. What are mechanical and electromagnetic waves?</h3>
<p>Mechanical waves require a medium for production and propagation — e.g., water and string waves. Electromagnetic waves need no medium — e.g., radio waves, heat and light waves, X-rays.</p>
<h3>Q6. Define crest, trough, wavelength and frequency.</h3>
<p>Crest: the raised part of a transverse wave, above the normal position. Trough: the dipped part, below the normal position. Wavelength (λ): distance between two consecutive crests (or troughs). Frequency (f): number of waves passing a point per second, measured in hertz.</p>
<h3>Q7. On what factors does the time period of a simple pendulum depend?</h3>
<p>It depends on the length <em>l</em> of the pendulum and the gravitational acceleration <em>g</em> (T = 2π√(l/g)), but it is independent of the mass of the bob.</p>
<h3>Q8. How is the wave equation v = fλ established with a ripple tank?</h3>
<p>A floating paper vibrates in place while waves pass; in one vibration (time T) the wave advances one wavelength, so λ = vT, giving v = λ/T = fλ.</p>
<h3>Q9. What is a stroboscope and how does it freeze waves?</h3>
<p>A rotating disc with edge slits. You see the wave only when a slit passes your eye; matching the slit rate to the wave's advance (one wavelength per slit interval) makes every glimpse identical, so the wave appears frozen.</p>
<h3>Q10. Define reflection, refraction, diffraction and interference.</h3>
<p>Reflection: waves bouncing back off a hurdle, obeying the laws of reflection. Refraction: bending of the wave's path when entering a different medium (speed and wavelength change; frequency stays the same). Diffraction: bending of waves around corners or obstacles — clearest when the slit size is close to the wavelength. Interference: the combined effect when two waves act on the medium's particles simultaneously (constructive when crest meets crest; destructive when crest meets trough).</p>
<h3>Q11. What are nodes and antinodes?</h3>
<p>In stationary waves, nodes are points of zero amplitude and antinodes are points of maximum amplitude. One wavelength equals twice the distance between two consecutive nodes.</p>
<h3>Q12. What is the fundamental frequency, and how do higher harmonics relate to it?</h3>
<p>The lowest frequency, at which the string makes a single loop, is the fundamental frequency (first harmonic). Higher harmonics are whole-number multiples: f_n = nf₁.</p>
<h3>Q13. Give an example showing energy is transferred through waves.</h3>
<p>Sound waves carry sound energy from one medium to another without any matter traveling along. Similarly, a cork floats and bobs when water waves reach it — it receives energy though the water itself does not move forward.</p>
<h3>Q14. What is amplitude, vibration and periodic motion?</h3>
<p>Amplitude is the maximum displacement from mean to extreme position. One vibration is a complete round trip about the mean position. Periodic motion repeats after regular time intervals.</p>
<h2>MCQs</h2>
<ol>
<li>The disturbance traveling through a medium is called: (a) wave motion (b) harmonic motion (c) random motion (d) both a and b — <strong>Answer: (a)</strong></li>
<li>The waves used to detect broken bones are: (a) light waves (b) X-rays (c) sound waves (d) both b and c — <strong>Answer: (b)</strong></li>
<li>If there is no extension in a spring, its position is called: (a) equilibrium (b) non-equilibrium (c) neutral equilibrium (d) stable equilibrium — <strong>Answer: (a)</strong></li>
<li>The unit of the spring constant is: (a) m (b) kg (c) N (d) N/m — <strong>Answer: (d)</strong></li>
<li>The restoring force on a mass <em>m</em> displaced by <em>x</em> on a spring is: (a) F = ma (b) F = kx (c) F = −kx (d) F = mg — <strong>Answer: (c)</strong></li>
<li>The ratio of applied force to displacement in a spring is called: (a) Hooke's law (b) spring constant (c) restoring force (d) amplitude — <strong>Answer: (b)</strong></li>
<li>The time for one complete round trip about the mean position is called: (a) time period (b) frequency (c) amplitude (d) wavelength — <strong>Answer: (a)</strong></li>
<li>The time period of a mass attached to a spring is: (a) T = 2π√(k/m) (b) T = 2π√(m/k) (c) T = 2π√(g/l) (d) T = 2π√(l/g) — <strong>Answer: (b)</strong></li>
<li>The time period of a simple pendulum is: (a) T = 2π√(l/g) (b) T = 2π√(m/k) (c) T = 2π√(g/l) (d) T = π√(l/g) — <strong>Answer: (a)</strong></li>
<li>Maximum displacement from the mean position is called: (a) wavelength (b) time period (c) amplitude (d) frequency — <strong>Answer: (c)</strong></li>
<li>In SHM, acceleration of the body is ______ proportional to displacement: (a) inversely (b) directly (c) equally (d) not — <strong>Answer: (b)</strong></li>
<li>At the mean position in SHM, acceleration is: (a) maximum (b) zero (c) 10 N (d) minimum — <strong>Answer: (b)</strong></li>
<li>At the mean position, kinetic energy of the pendulum bob is: (a) minimum (b) less (c) maximum (d) zero — <strong>Answer: (c)</strong></li>
<li>At the extreme position, potential energy of the pendulum is: (a) maximum (b) minimum (c) zero (d) both a and b — <strong>Answer: (a)</strong></li>
<li>Waves in which medium particles vibrate parallel to wave direction are: (a) longitudinal waves (b) transverse waves (c) electromagnetic waves (d) both b and c — <strong>Answer: (a)</strong></li>
<li>Waves in which medium particles vibrate perpendicular to wave direction are: (a) electromagnetic waves (b) mechanical waves (c) both a and b (d) transverse waves — <strong>Answer: (d)</strong></li>
<li>Sound waves are an example of: (a) longitudinal waves (b) transverse waves (c) electromagnetic waves (d) none of these — <strong>Answer: (a)</strong></li>
<li>Energy can be transferred from one place to another: (a) through matter (b) through waves (c) both a and b (d) none of these — <strong>Answer: (c)</strong></li>
<li>Waves show the properties: (a) reflection (b) refraction (c) diffraction (d) all of these — <strong>Answer: (d)</strong></li>
<li>The instrument used to freeze moving waves is: (a) ripple tank (b) vibrator (c) stroboscope (d) pendulum — <strong>Answer: (c)</strong></li>
<li>The instrument used to study wave properties is: (a) ripple tank (b) stroboscope (c) pendulum (d) spring — <strong>Answer: (a)</strong></li>
<li>The part of a wave below the normal position is called: (a) extreme position (b) crest (c) trough (d) node — <strong>Answer: (c)</strong></li>
<li>Distance between two consecutive crests is called: (a) wavelength (b) frequency (c) time period (d) amplitude — <strong>Answer: (a)</strong></li>
<li>The number of waves passing a point in one second is: (a) time period (b) cycle (c) frequency (d) wavelength — <strong>Answer: (c)</strong></li>
<li>The unit of frequency is: (a) hertz (b) vibrations per second (c) cycles per second (d) all of these — <strong>Answer: (d)</strong></li>
<li>Wave speed is calculated by: (a) vT (b) S/T (c) fλ (d) Tf — <strong>Answer: (c)</strong></li>
<li>Water waves after striking a hurdle will: (a) reflect (b) refract (c) diffract (d) all of a, b, c — <strong>Answer: (a)</strong></li>
<li>The wavelength of a stationary wave is how many times the distance between two consecutive nodes? (a) three times (b) four times (c) two times (d) five times — <strong>Answer: (c)</strong></li>
<li>The part of a stationary wave with maximum amplitude is: (a) antinode (b) node (c) crest (d) trough — <strong>Answer: (a)</strong></li>
<li>Interference of waves is of two types: (a) constructive (b) destructive (c) both a and b (d) none of these — <strong>Answer: (c)</strong></li>
<li>Time period is the reciprocal of: (a) frequency (b) cycle (c) wavelength (d) amplitude — <strong>Answer: (a)</strong></li>
<li>The product of frequency and time period equals: (a) v (b) 1 (c) 9 (d) λ — <strong>Answer: (b)</strong></li>
<li>If the mass of a simple pendulum's bob is doubled, its time period: (a) is doubled (b) becomes four times (c) remains the same (d) is halved — <strong>Answer: (c)</strong></li>
<li>If the length of a simple pendulum is halved, its time period becomes: (a) T/√2 (b) T/2 (c) 2T (d) √2·T — <strong>Answer: (a)</strong></li>
<li>In terms of wavelength, the distance between a node and the neighbouring antinode is: (a) λ/4 (b) λ/2 (c) λ (d) 2λ — <strong>Answer: (a)</strong></li>
</ol>
<h2>Key Formulas & Definitions</h2>
<table>
<tr><th>Quantity</th><th>Formula / Definition</th></tr>
<tr><td>Hooke's law</td><td>F = kx</td></tr>
<tr><td>Spring constant</td><td>k = F/x (unit: N/m)</td></tr>
<tr><td>Restoring force</td><td>F = −kx</td></tr>
<tr><td>SHM condition</td><td>a = −(k/m)x — acceleration ∝ displacement, toward mean position</td></tr>
<tr><td>Time period (mass–spring)</td><td>T = 2π√(m/k)</td></tr>
<tr><td>Time period (simple pendulum)</td><td>T = 2π√(l/g)</td></tr>
<tr><td>Wave equation</td><td>v = fλ</td></tr>
<tr><td>Frequency–period relation</td><td>f = 1/T (so f × T = 1)</td></tr>
<tr><td>Stationary wave harmonics</td><td>f_n = nf₁</td></tr>
<tr><td>Stationary wavelength</td><td>λ = 2 × (distance between consecutive nodes)</td></tr>
<tr><td>Law of reflection</td><td>Angle of incidence = angle of reflection</td></tr>
</table>
<h2>Solved Numericals</h2>
<h3>Numerical 1 — Pendulum on Earth and Moon</h3>
<p><strong>Question:</strong> Find the time period of a 1-metre simple pendulum on Earth and on the Moon, where g is one-sixth of Earth's value (take g = 10 m/s²).</p>
<p><strong>Solution:</strong> l = 1 m; on Earth g = 10 m/s², on the Moon g = 10/6 ≈ 1.67 m/s². Using T = 2π√(l/g):<br>
On Earth: T = 2 × 3.14 × √(1/10) = 6.28 × 0.316 ≈ <strong>1.99 s</strong>.<br>
On the Moon: T = 2 × 3.14 × √(1/1.67) = 6.28 × 0.774 ≈ <strong>4.86 s</strong>.</p>
<h3>Numerical 2 — Mass on a Spring</h3>
<p><strong>Question:</strong> A 0.5 kg body is attached to a spring (k = 8 N/m) on a frictionless horizontal surface. Find its time period.</p>
<p><strong>Solution:</strong> T = 2π√(m/k) = 2 × 3.14 × √(0.5/8) = 6.28 × √(0.0625) = 6.28 × 0.25 = <strong>1.57 s</strong>.</p>
<h3>Numerical 3 — Wave Frequency from Wavelength and Speed</h3>
<p><strong>Question:</strong> A wave of wavelength 0.1 m travels at 2 m/s. Find its frequency.</p>
<p><strong>Solution:</strong> f = v/λ = 2/0.1 = <strong>20 Hz</strong>.</p>
<h3>Numerical 4 — Stationary Wave Harmonics</h3>
<p><strong>Question:</strong> A stationary wave has fundamental frequency 250 Hz. Find the next three higher frequencies.</p>
<p><strong>Solution:</strong> f_n = nf₁ → f₂ = 2 × 250 = <strong>500 Hz</strong>; f₃ = 3 × 250 = <strong>750 Hz</strong>; f₄ = 4 × 250 = <strong>1000 Hz</strong>.</p>
<h3>Numerical 5 — Pendulum Length for a 2-Second Period</h3>
<p><strong>Question:</strong> A simple pendulum has time period 2 s. What is its length on Earth (g = 10 m/s²) and on the Moon (g = 1.67 m/s²)?</p>
<p><strong>Solution:</strong> From T = 2π√(l/g), l = gT²/(4π²). With T = 2 s: l = 4g/39.48.<br>
Earth: l = 40/39.48 ≈ <strong>1.01 m</strong>.<br>
Moon: l = (4 × 1.67)/39.48 = 6.68/39.48 ≈ <strong>0.17 m</strong>.</p>
<h3>Numerical 6 — Spring Constant</h3>
<p><strong>Question:</strong> A spring of natural length 8 cm stretches to 16 cm when a 4 kg mass hangs from it. Find the spring constant.</p>
<p><strong>Solution:</strong> Extension x = 16 − 8 = 8 cm = 0.08 m. Force F = mg = 4 × 10 = 40 N. k = F/x = 40/0.08 = <strong>500 N/m</strong>.</p>
<h3>Numerical 7 — Frequency, Period and Speed</h3>
<p><strong>Question:</strong> Five waves pass a point in 10 s. Find the frequency and time period; if the wavelength is 5 cm, find the wave speed.</p>
<p><strong>Solution:</strong> f = 5/10 = <strong>0.5 Hz</strong>; T = 1/f = <strong>2 s</strong>; v = fλ = 0.5 × 0.05 = <strong>0.025 m/s</strong>.</p>
<h3>Numerical 8 — Wavelength from Period and Speed</h3>
<p><strong>Question:</strong> A wave has time period 5 s and speed 10 m/s. Find its wavelength.</p>
<p><strong>Solution:</strong> λ = vT = 10 × 5 = <strong>50 m</strong>.</p>
<h3>Numerical 9 — Fundamental Frequency of a Stationary Wave</h3>
<p><strong>Question:</strong> A stationary wave makes 5 loops. The distance between two consecutive nodes is 10 cm and the wave speed is 20 m/s. Find the fundamental frequency.</p>
<p><strong>Solution:</strong> λ = 2 × 0.10 = 0.20 m. f₅ = v/λ = 20/0.2 = 100 Hz. Since f₅ = 5f₁, f₁ = 100/5 = <strong>20 Hz</strong>.</p>
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