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Unit 10 SIMPLE HARMONIC MOTION AND WAVES After studying this unit, students will be able to: state the conditions necessary for an object to oscillate with SHM. explain SHM with simple pendulum, ball and bowl examples. draw forces acting on a displaced pendulum....

Unit 10 SIMPLE HARMONIC MOTION AND WAVES After studying this unit, students will be able to: state the conditions necessary for an object to oscillate with SHM. explain SHM with simple pendulum, ball and bowl examples. draw forces acting on a displaced pendulum. solve problems by using the formula T = 2π ll l /g g for simple pendulum. understand that damping progressively reduces the amplitude of oscillation. describe wave motion as illustrated by vibrations in rope, slinky spring and by experiments with water waves. describe that waves are means of energy transfer without transfer of matter. distinguish between mechanical and electromagnetic waves. identify transverse and longitudinal waves in mechanical media, slinky and springs. define the terms speed (v), frequency ( f ), wavelength (λ), time period (T ), amplitude, crest, trough, cycle, wavefront, compression and rarefaction. derive equation v = f λ. solve problems by applying the relation f = 1/T and v = f λ. describe properties of waves such as reflection, refraction and diffraction with the help of ripple tank. Science, Technology and Society Connections The students will be able to: explain the diffraction of radiowaves but not of T.V waves (transmission can be heard in such areas where the waves cannot reach directly). SIMPLE HARMONIC MOTION AND WAVES A body is said to be vibrating if it moves back and forth or to For your information and fro about a point. Another term for vibration is oscillation. A special kind of vibratory or oscillatory motion is called the simple harmonic motion (SHM), which is the main focus of this chapter. We will discuss important characteristics of SHM and systems executing SHM. We will also introduce different types of waves and will demonstrate their properties with the help of ripple tank. A spider detects its prey due to vibration produced in the web. 10.1 SIMPLE HARMONIC MOTION (SHM) In the following sections we will discuss simple harmonic motion of different systems. The motion of mass attached to a spring on a horizontal frictionless surface, the motion of a ball placed in a bowl and the motion of a bob attached to a string are examples of SHM. MOTION OF MASS ATTACHED TO A SPRING One of the simplest types of oscillatory motion is that of horizontal mass-spring system (Fig.10.1). If the spring is stretched or compressed through a small displacement x from its mean position, it exerts a force F on the mass. F=0 According to Hooke’s law this force is directly proportional to the change in length x of the spring i.e., x (a) B O A F=-kx........ (10.1) x=0 F where x is the displacement of the mass from its mean position O, and k is a constant called the spring constant defined as (b) k=- F x x The value of k is a measure of the stiffness of the spring. Stiff F springs have large value of k and soft springs have small value of k. (c) x As F = ma B O A Therefore, k = - ma Fig.10.1: SHM of a mass-spring x system or a=- k x m a  - x........ (10.2) It means that the acceleration of a mass attached to a spring is directly proportional to its displacement from the mean position. Hence, the horizontal motion of a mass-spring system is an example of simple harmonic motion. Not For Sale – PESRP 2 SIMPLE HARMONIC MOTION AND WAVES The negative sign in Eq. 10.1 means that the force exerted by the spring is always directed opposite to the displacement of the mass. Because the spring force always acts towards the mean position, it is sometimes called a restoring force. A restoring force always pushes or pulls the object performing oscillatory motion towards the mean position. For your information Initially the mass m is at rest in mean position O and the x = -A x = 0 x = +A resultant force on the mass is zero (Fig.10.1-a). Suppose the mass is pulled through a distance x up to extreme position A and then released (Fig.10.1-b). The restoring force exerted by the spring on the mass will pull it K.E =0 K.E = max P.E = max K.E = 0 towards the mean position O. Due to the restoring force P.E = 0 P.E = max the mass moves back, towards the mean position O. The Kinetic and potential energy at magnitude of the restoring force decreases with the different positions in a distance from the mean position and becomes zero at O. mass–spring system. However, the mass gains speed as it moves towards the mean position and its speed becomes maximum at O. Due to inertia the mass does not stop at the mean position O but continues its motion and reaches the extreme position B. As the mass moves from the mean position O to the extreme position B, the restoring force acting on it towards the mean position steadily increases in strength. Hence the speed of the mass decreases as it moves towards the extreme position Tidbits B. The mass finally comes briefly to rest at the extreme A human eardrum can oscillate position B (Fig. 10.1-c). Ultimately the mass returns to the back and forth up to 20,000 mean position due to the restoring force. times in one second. This process is repeated, and the mass continues to oscillate back and forth about the mean position O. Such motion of a mass attached to a spring on a horizontal frictionless surface is known as Simple Harmonic Motion (SHM). Quick Quiz The time period T of the simple harmonic motion of a mass What is the displacement of an ‘m’ attached to a spring is given by the following equation: object in SHM when the kinetic and potential energies are m......... (10.3) equal? T  2 k´ 3 Not For Sale – PESRP SIMPLE HARMONIC MOTION AND WAVES BALL AND BOWL SYSTEM The motion of a ball placed in a bowl is another example of B Ball A simple harmonic motion (Fig 10.2). When the ball is at the R mean position O, that is, at the centre of the bowl, net force acting on the ball is zero. In this position, weight of the ball acts downward and is equal to the upward normal force of O Bowl the surface of the bowl. Hence there is no motion. Now if we w = mg bring the ball to position A and then release it, the ball will Fig. 10.2: When a ball is gently start moving towards the mean position O due to the displaced from the centre of a restoring force caused by its weight. At position O the ball bowl it starts oscillating about gets maximum speed and due to inertia it moves towards the the centre due to force of gravity which acts as a extreme position B. While going towards the position B, the restoring force speed of the ball decreases due to the restoring force which acts towards the mean position. At the position B, the ball stops for a while and then again moves towards the mean position O under the action of the restoring force. This to and fro motion of the ball continues about the mean position O till all its energy is lost due to friction. Thus the to and fro motion of the ball about a mean position placed in a bowl is an example of simple harmonic motion. MOTION OF A SIMPLE PENDULUM A simple pendulum also exhibits SHM. It consists of a small θ bob of mass ‘m’ suspended from a light string of length ‘ l ’ T T fixed at its upper end. In the equilibrium position O, the net l force on the bob is zero and the bob is stationary. Now if we T B bring the bob to extreme position A, the net force is not zero A m S m (Fig.10.3). There is no force acting along the string as the o m mgsinθ tension in the string cancels the component of the weight mg mgcosθ mg cos θ. Hence there is no motion along this direction. w = mg The component of the weight mg sin θ is directed towards the Mean position mean position and acts as a restoring force. Due to this force Fig. 10.3: Forces acting on a the bob starts moving towards the mean position O. At O, the displaced pendulum. The bob has got the maximum velocity and due to inertia, it does restoring force that causes the not stop at O rather it continues to move towards the pendulum to undergo simple extreme position B. During its motion towards point B, the harmonic motion is the component of gravitational velocity of the bob decreases due to restoring force. The force mg sinθ tangent to the velocity of the bob becomes zero as it reaches the point B. path of motion Not For Sale – PESRP 4 SIMPLE HARMONIC MOTION AND WAVES The restoring force mgsinθ still acts towards the mean position O and due to this force the bob again starts moving towards the mean position O. In this way, the bob continues its to and fro motion about the mean position O. Time Period It is clear from the above discussion that the speed of the bob increases while moving from point A to O due to the restoring force which acts towards O. Therefore, acceleration of the bob is also directed towards O. Similarly, when the bob moves from O to B, its speed decreases due to restoring force which again acts towards O. Therefore, acceleration of the bob is again directed towards O. It follows that the acceleration of the bob is always directed towards the mean position O. Hence the motion of a simple pendulum is SHM. We have the following formula for the time period of a simple pendulum Time period of a pendulum is the time to complete one ll......... (10.4) cycle. T  2 ß g From the motion of these simple systems, we can define SHM as: For your information Simple harmonic motion occurs when the net force is The period of a pendulum is directly proportional to the displacement from the mean independent of its mass and position and is always directed towards the mean amplitude. position. In other words, when an object oscillates about a fixed position (mean position) such that its acceleration is directly proportional to its displacement from the mean position and is always directed towards the mean position, its motion is Check Your Understanding called SHM. Tell whether or not these motions are examples of simple harmonic motion: Important features of SHM are summarized as: (a) up and down motion of a i. A body executing SHM always vibrates about a fixed leaf in water pond (b) motion position. of a ceiling fan (c) motion of ii. Its acceleration is always directed towards the mean hands of clock (d) motion of a position. plucked string fixed at both its ends (e) movement of honey iii. The magnitude of acceleration is always directly bee. proportional to its displacement from the mean 5 Not For Sale – PESRP SIMPLE HARMONIC MOTION AND WAVES position i.e., acceleration will be zero at the mean position while it will be maximum at the extreme positions. iv. Its velocity is maximum at the mean position and zero at the extreme positions. Now we discuss different terms which characterize simple harmonic motion. Vibration: One complete round trip of a vibrating body about its mean position is called one vibration. Time Period (T ): The time taken by a vibrating body to complete one vibration is called time period. Frequency ( f ): The number of vibrations or cycles of a For your information vibrating body in one second is called its frequency. It is reciprocal of time period i.e., f = 1/T Amplitude (A): The maximum displacement of a vibrating body on either side from its mean position is called its amplitude. Example 10.1: Find the time period and frequency of a simple -2 pendulum 1.0 m long at a location where g = 10.0 m s. Christian Huygens invented Solution: Given, l = 1.0 m, g = 10.0 m s. -2 the pendulum clock in 1656. He was inspired by the work of Using the formula, Galileo who had discovered T  2 l gßg that all pendulums of the same length took the same amount By putting the values of time to complete one full swing. Huygens developed the 1.0 m first clock that could accurately T  2 3.14  T = 1.99 s 10.0 m ss2 measure time. Frequency of simple pendulum is given by f = 1/T = 1/1.99 s = 0.50 Hz 10.2 DAMPED OSCILLATIONS Vibratory motion of ideal systems in the absence of any friction or resistance continues indefinitely under the action of a restoring force. Practically, in all systems, the force of friction retards the motion, so the systems do not oscillate indefinitely. The friction reduces the mechanical energy of Not For Sale – PESRP 6 SIMPLE HARMONIC MOTION AND WAVES the system as time passes, and the motion is said to be Displacement + Decreasing damped. This damping progressively reduces the amplitude amplitide of the vibration of motion as shown in Fig. 10.4. o Time Shock absorbers in automobiles are one practical application “Envelope” of of damped motion. A shock absorber consists of a piston - the damping Fig. 10.4: The variation of moving through a liquid such as oil (Fig.10.5). The upper part amplitude with time of of the shock absorber is firmly attached to the body of the car. damping system When the car travels over a bump on the road, the car may vibrate violently. The shock absorbers damp these vibrations and convert their energy into heat energy of the oil. Thus Attached to car frame The oscillations of a system in the presence of some resistive force are damped oscillations. Piston Liquid 10.3 WAVE MOTION Attached to car axle Waves play an important role in our daily life. It is because Fig. 10.5: Shock absorber waves are carrier of energy and information over large distances. Waves require some oscillating or vibrating source. Pencil Here we demonstrate the production and propagation of different waves with the help of vibratory motion of objects. Activity 10.1: Dip one end of a pencil into a tub of water, and move it up and down vertically (Fig. 10.6). The disturbance in Cork the form of ripples produces water waves, which move away from the source. When the wave reaches a small piece of cork floating near the disturbance, it moves up and down about its Fig. 10.6: Waves produced by dipping a pencil in a water tub original position while the wave will travel outwards. The net displacement of the cork is zero. The cork repeats its Support P vibratory motion about its mean position. Crest Activity 10.2: Take a rope and mark a point P on it. Tie one P end of the rope with a support and stretch the rope by P holding its other end in your hand (Fig. 10.7). Now, flipping the rope up and down regularly will set up a wave in the rope which will travel towards the fixed end. The point P on the rope will start vibrating up and down as the wave passes P Trough across it. The motion of point P will be perpendicular to the Fig. 10.7: Waves produced in a direction of the motion of wave. rope 7 Not For Sale – PESRP SIMPLE HARMONIC MOTION AND WAVES From the above simple activities, we can define wave as: For your information A wave is a disturbance in the medium which causes Electric field the particles of the medium to undergo vibratory Magnetic field motion about their mean position in equal intervals of time. There are two categories of waves: Direction of wave motion 1. Mechanical waves Electromagnetic waves consist 2. Electromagnetic waves of electric and magnetic fields oscillating perpendicular to Mechanical Waves: Waves which require any medium for each other. their propagation are called mechanical waves. Examples of mechanical waves are water waves, sound waves and waves produced on the strings and springs. Quick Quiz Do mechanical waves pass Electromagnetic Waves: Waves which do not require any through vacuum, that is, empty space? medium for their propagation are called electromagnetic waves. Radiowaves, television waves, X-rays, heat and light waves are some examples of electromagnetic waves. 10.4 TYPES OF MECHANICAL WAVES Depending upon the direction of displacement of medium with respect to the direction of the propagation of wave itself, mechanical waves may be classified as longitudinal or transverse. Longitudinal waves can be produced on a spring (slinky) placed on a smooth floor or a long bench. Fix one end of the slinky with a rigid support and hold the other end into your hand. Now give it a regular push and pull quickly in the direction of its length (Fig.10.8). Not For Sale – PESRP 8 SIMPLE HARMONIC MOTION AND WAVES Movement of hand Support Direction of wave travel Displacement of particles Compression λ For your Information Compression Rarefaction Compression Longitudinal waves move Fig. 10.8: Longitudinal wave on a slinky Fig. 10.8: Longitudinal wave on a slinky faster through solids than through gases or liquids. A series of disturbances in the form of waves will start moving Tra n sve rs e wave s m o ve along the length of the slinky. Such a wave consists of regions through solids at a speed of called compressions, where the loops of the spring are close less than half of the speed of together, alternating with regions called rarefactions longitudinal waves. It is (expansions), where the loops are spaced apart. In the regions of because the restoring force exerted during this up and compression, particles of the medium are closer together while down motion of particles of in the regions of rarefaction, particles of the medium are spaced the medium is less than the apart. The distance between two consecutive compressions is restoring force exerted by a called wavelength. The compressions and rarefactions move back and forth motion of back and forth along the direction of motion of the wave. Such a particles of the medium in case of longitudinal waves. wave is called longitudinal wave and is defined as: In longitudinal waves the particles of the medium move back and forth along the direction of propagation of wave. We can produce transverse waves with the help of a slinky. Stretch out a slinky along a smooth floor with one end fixed. Grasp the other end of the slinky and move it up and down quickly (Fig.10.9). A wave in the form of alternate crests and troughs will start travelling towards the fixed end. The crests are the highest points while the troughs are the lowest points of the particles of the medium from the mean position. The distance between two consecutive crests or troughs is called 9 Not For Sale – PESRP SIMPLE HARMONIC MOTION AND WAVES wavelength. The crests and troughs move perpendicular to the direction of the wave. λ Wave movement Crest Particle movement Movement of hand Support from side to side Troughs Fig. 10.9: Transverse wave on a slinky Therefore, transverse waves can be defined as: In case of transverse waves, the vibratory motion of particles of the medium is perpendicular to the direction of propagation of waves. Waves on the surface of water and light waves are examples of transverse waves. WAVES AS CARRIERS OF ENERGY Energy can be transferred from one place to another through waves. For example, when we shake the stretched string up and down, we provide our muscular energy to the string. As a result, a set of waves can be seen travelling along the string. The vibrating force from the hand disturbs the particles of the string and sets them in motion. These particles then transfer their energy to the adjacent particles in the string. Energy is thus transferred from one place of the medium to the other in the form of wave. The amount of energy carried by the wave depends on the distance of the stretched string from its rest position. That is, the energy in a wave depends on the amplitude of the wave. If we shake the string faster, we give more energy per second to produce wave of higher frequency, and the wave delivers more energy per second to the particles of the string as it moves forward. Water waves also transfer energy from one place to another Not For Sale – PESRP 10 SIMPLE HARMONIC MOTION AND WAVES as explained below: Activity 10.3: Drop a stone into a pond of water. Water waves will be produced on the surface of water and will travel outwards (Fig. 10.10). Place a cork at some distance from the falling stone. When waves reach the cork, it will move up and down alongwith the motion of the water particles by getting For your information energy from the waves. Generating a high frequency wave, requires more energy per second than to generate a low frequency wave. Thus, a high frequency wave carries more energy than a low frequency wave of the same amplitude. Cork and water go up and down Energy travels in this direction Fig. 10.10 This activity shows that water waves like other waves transfer energy from one place to another without transferring matter, i.e., water. RELATION BETWEEN VELOCITY, FREQUENCY AND Do you know? WAVELENGTH Earthquake produces waves Wave is a disturbance in a medium which travels from one through the crust of the Earth place to another and hence has a specific velocity of travelling. in the form of seismic waves. By studying such waves, the This is called the velocity of wave which is defined by geophysicists learn about the Velocity = distance/time internal structure of the Earth v = d and information about the t occurrence of future Earth If time taken by the wave in moving from one point to another activity. is equal to its time period T, then the distance covered by the wave will be equal to one wavelength λ, hence we can write: v = λ T 1 But time period T, is reciprocal of the frequency f, i.e., T  f 11 Not For Sale – PESRP SIMPLE HARMONIC MOTION AND WAVES Therefore, v =f λ......... (10.5) Eq. (10.5) is true both for longitudinal and transverse waves. Example 10.2: A wave moves on a slinky with frequency of 4 Hz and wavelength of 0.4 m. What is the speed of the wave? Solution: Given that, f = 4 Hz, λ = 0.4 m Wave speed v =f λ = (4 Hz) (0.4 m) v = 1.6 m s-1 10.5 RIPPLE TANK Ripple tank is a device to produce water waves and to study their characteristics. This apparatus consists of a rectangular tray having glass bottom and is placed nearly half metre above the surface of a table (Fig. 10.11). Waves can be produced on the surface of water present in the tray by means of a vibrator (paddle). Lamp Power supply Shallow tank of water Oscillating paddle Wave patterns on a viewing screen Fig. 10.11: Ripple tank apparatus λ This vibrator is an oscillating electric motor fixed on a wooden plate over the tray such that its lower surface just touches the surface of water. On setting the vibrator ON, this wooden plate starts vibrating to generate water waves consisting of straight wavefronts (Fig.10.12). An electric bulb is hung above the tray to observe the image of water waves on the Fig. 10.12: Waves consisting of paper or screen. The crests and troughs of the waves appear straight wavefronts as bright and dark lines respectively, on the screen. Now we explain the reflection of water waves with the help of ripple tank. Not For Sale – PESRP 12 SIMPLE HARMONIC MOTION AND WAVES Place a barrier in the ripple tank. The water waves will reflect Quick Quiz from the barrier. If the barrier is placed at an angle to the What do the dark and bright fringes on the screen of ripple wavefront, the reflected waves can be seen to obey the law of tank represent? reflection i.e., the angle of the incident wave along the normal will be equal to the angle of the reflected wave Angle of (Fig.10.13). Thus, we define reflection of waves as: incidence When waves moving in one medium fall on the surface of Normal another medium they bounce back into the first medium such Incident i Barrier waves that the angle of incidence is equal to the angle of reflection. (a) The speed of a wave in water depends on the depth of water. If a block is submerged in the ripple tank, the depth of water in the tank will be shallower over the block than elsewhere. Normal Angle of When water waves enter the region of shallow water their reflection r wavelength decreases (Fig.10.14). But the frequency of the water waves remains the same in both parts of water Barrier because it is equal to the frequency of the vibrator. (b) λ2 Shallow water Reflected waves Boundary between (slow speed) Fig. 10.13: Reflection of water v2 waves from a plane barrier deep and Wavefront shallow water Deep water Straight wave λ1 v1 (faster speed) generator Ripple tank Fig. 10.14 For the observation of refraction of water waves, we repeat the above experiment such that the boundary between the deep and the shallower water is at some angle to the wavefront (Fig. 10.15). Now we will observe i that in addition to the change in wavelength, the waves change their direction of propagation as well. Note that i r the direction of propagation is always normal to the wavefronts. This change of path of water waves while r passing from a region of deep water to that of shallower Fig. 10.15: Refraction of water one is called refraction which is defined as: waves 13 Not For Sale – PESRP SIMPLE HARMONIC MOTION AND WAVES When a wave from one medium enters into the second medium at some angle, its direction of travel changes. Now we observe the phenomenon of diffraction of water waves. Generate straight waves in a ripple tank and place two obstacles in line in such a way that separation between them is equal to the wavelength of water waves. After passing through a small slit between the two obstacles, the waves will spread in every direction and change into almost semicircular pattern (Fig. 10.16). Diffraction of waves can only be observed clearly if the size of the obstacle is comparable with the wavelength of the wave. Fig.10.16: Diffraction of water Fig.10.17 shows the diffraction of waves while passing through waves through a small slit a slit with size larger than the wavelength of the wave. Only a small diffraction occurs near the corners of the obstacle. The bending or spreading of waves around the sharp edges or corners of obstacles or slits is called diffraction. Example 10.3: A student performs an experiment with waves in water. The student measures the wavelength of a wave to be 10 cm. By using a stopwatch and observing the oscillations Fig.10.17: Diffraction of water of a floating ball, the student measures a frequency of 2 Hz. If waves through a large slit the student starts a wave in one part of a tank of water, how long will it take the wave to reach the opposite side of the Deep tank 2 m away? Solution: Shallow (1) We are given the frequency, wavelength, and distance. (2) We have to calculate the time, the wave takes to move a Fig.10.18 distance of 2 m. (3) The relationship between frequency, wavelength, and speed is ACTIVITY v = f λ. The relationship between time, speed, and distance is Study Fig. 10.18 to answer the v = d /t following questions: (4) Rearrange the speed formula to solve for the time: t = d / v 1. What happens to the direction of wave when water The speed of the wave is the frequency times the wavelength. waves pass from deep to v = f λ = (2 Hz)(0.1 m) = 0.2 m s-1. shallow part of the water? Use this value to calculate the time: 2. Are the magnitudes of angle t = 2 m/0.2 m s-1 = 10 s of incidence and angle of refraction equal? 3. Which will be greater? Not For Sale – PESRP 14 SIMPLE HARMONIC MOTION AND WAVES SUMMARY  Simple harmonic motion (SHM) is a to and fro oscillatory motion in which acceleration of the body is directly proportional to the displacement of the body from the mean position and is always directed towards the mean position.  The motion of a mass attached to a spring, simple pendulum and that of a ball inside a bowl is SHM.  Time taken by the simple pendulum to complete one cycle is called its time period. It depends upon the length of the pendulum and is independent of the mass and amplitude of the pendulum.  The number of cycles completed in one second is called frequency of a vibrating body. It is reciprocal of time period.  The maximum displacement from mean position of a body performing SHM is called amplitude.  Wave is a phenomenon of transferring energy from one place to another without the transfer of matter.  Mechanical waves are those waves which require some medium for their propagation.  Electromagnetic waves do not require any medium for their propagation.  Transverse waves are the mechanical waves in which particles of the medium vibrate about their mean position perpendicular to the direction of propagation of the waves.  Compressional (longitudinal) waves are the mechanical waves in which particles of the medium vibrate about their mean position along the direction of propagation of the waves.  The speed (v) of a wave is equal to the product of frequency ( f ) and wavelength (λ) i.e.,v =f λ.  Ripple tank is a device used to produce water waves and to demonstrate different properties of water waves like reflection, refraction and diffraction.  When a wave travelling from one medium falls on the surface of another medium, it may bounce back into the first medium. This phenomenon is called reflection of waves.  When waves from one medium enter the second medium at some angle their direction of travel may change. This phenomenon is called refraction of waves. The speed and wavelength of wave change in different media but frequency does not change.  The bending of waves around obstacles or sharp edges is called diffraction of waves. 15 Not For Sale – PESRP SIMPLE HARMONIC MOTION AND WAVES MULTIPLE CHOICE QUESTIONS Choose the correct answer from the following choices: i. Which of the following is an example of simple harmonic motion? (a) the motion of simple pendulum (b) the motion of ceiling fan (c) the spinning of the Earth on its axis (d) a bouncing ball on a floor ii. If the mass of the bob of a pendulum is increased by a factor of 3, the period of the pendulum's motion will (a) be increased by a factor of 2 (b) remain the same (c) be decreased by a factor of 2 (d) be decreased by a factor of 4 iii. Which of the following devices can be used to produce both transverse and longitudinal waves? (a) a string (b) a ripple tank (c) a helical spring (slinky) (d) a tuning fork iv. Waves transfer (a) energy (b) frequency (c) wavelength (d) velocity v. Which of the following is a method of energy transfer? (a) conduction (b) radiation (c) wave motion (d) all of these vi. In a vacuum, all electromagnetic waves have the same (a) speed (b) frequency (c) amplitude (d) wavelength vii. A large ripple tank with a vibrator working at a frequency of 30 Hz produces 25 complete waves in a distance of 50 cm. The velocity of the wave is -1 -1 (a) 53 cm s (b) 60 cm s -1 -1 (c) 750 cm s (d) 1500 cm s viii. Which of the following characteristics of a wave is independent of the others? (a) speed (b) frequency (c) amplitude (d) wavelength ix. The relation between v, f and λ of a wave is (a) v f = λ (b) f λ = v (c) v λ = f (d) v = λ / f Not For Sale – PESRP 16 SIMPLE HARMONIC MOTION AND WAVES REVIEW QUESTIONS 10.1. What is simple harmonic motion? What are the necessary conditions for a body to execute simple harmonic motion? 10.2. Think of several examples of motion in everyday life that are simple harmonic. 10.3. What are damped oscillations. How damping progressively reduces the amplitude of oscillation? 10.4. How can you define the term wave? Elaborate the difference between mechanical and electromagnetic waves. Give examples of each. 10.5. Distinguish between longitudinal and transverse waves with suitable examples. 10.6. Draw a transverse wave with an amplitude of 2 cm and a wavelength of 4 cm. Label a crest and trough on the wave. 10.7. Derive a relationship between velocity, frequency and wavelength of a wave. Write a formula relating velocity of a wave to its time period and wavelength. 10.8. Waves are the means of energy transfer without transfer of matter. Justify this statement with the help of a simple experiment. 10.9. Explain the following properties of waves with reference to ripple tank experiment: a. Reflection b. Refraction c. Diffraction 10.10. Does increasing the frequency of a wave also increase its wavelength? If not, how are these quantities related? CONCEPTUAL QUESTIONS 10.1. If the length of a simple pendulum is doubled, what will be the change in its time period? 10.2. A ball is dropped from a certain height onto the floor and keeps bouncing. Is the motion of the ball simple harmonic? Explain. 10.3. A student performed two experiments with a simple pendulum. He/She used two bobs of different masses by keeping other parameters constant. To his/her astonishment the time period of the pendulum did not change! Why? 10.4. What types of waves do not require any material medium for their propagation? 10.5. Plane waves in the ripple tank undergo refraction when they move from deep to shallow water. What change occurs in the speed of the waves? NUMERICAL PROBLEMS 10.1. The time period of a simple pendulum is 2 s. What will be its length on the Earth? -2 What will be its length on the Moon if gm =ge/6? where ge = 10 m s. Ans.(1.02 m, 0.17 m) 10.2. A pendulum of length 0.99 m is taken to the Moon by an astronaut. The period of the pendulum is 4.9 s. What is the value of g on the surface of the Moon? 17 Not For Sale – PESRP SIMPLE HARMONIC MOTION AND WAVES -2 Ans.(1.63 m s ) 10.3. Find the time periods of a simple pendulum of 1 metre length, placed on Earth and th on Moon. The value of g on the surface of Moon is 1/6 of its value on Earth, -2 where ge is 10 m s. Ans.(2 s, 4.9 s) 10.4. A simple pendulum completes one vibration in two seconds. Calculate its length, -2 when g = 10.0 m s. Ans. (1.02 m) 10.5. If 100 waves pass through a point of a medium in 20 seconds, what is the frequency and the time period of the wave? If its wavelength is 6 cm, calculate the wave speed. -1 Ans. (5 Hz, 0.2 s, 0.3 m s ) 10.6. A wooden bar vibrating into the water surface in a ripple tank has a frequency of 12 Hz. The resulting wave has a wavelength of 3 cm. What is the speed of the wave? -1 Ans. (0.36 m s ) 10.7. A transverse wave produced on a spring has a frequency of 190 Hz and travels along the length of the spring of 90 m, in 0.5 s. (a) What is the period of the wave? (b) What is the speed of the wave? (c) What is the wavelength of the wave? -1 Ans. (0.01 s, 180 m s , 0.95 m) 10.8. Water waves in a shallow dish are 6.0 cm long. At one point, the water moves up and down at a rate of 4.8 oscillations per second. (a) What is the speed of the water waves? (b) What is the period of the water waves? -1 Ans. (0.29 m s , 0.21 s) 10.9. At one end of a ripple tank 80 cm across, a 5 Hz vibrator produces waves whose wavelength is 40 mm. Find the time the waves need to cross the tank. Ans. (4 s) 10.10. What is the wavelength of the radiowaves transmitted by an FM station at 90 MHz? 6 8 -1 where 1M = 10 , and speed of radiowave is 3 x 10 m s. Ans. (3.33 m) Not For Sale – PESRP 18