CIRCULAR MOTION AND SHM: Solutions to Higher Level Questions

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1 CIRCULAR MOTION AND SHM: Solutions to Higher Level Questions ****ALL QUESTIONS-ANSWERS ARE HIGHER LEVEL*** Solutions 015 Question 6 (i) Explain what is meant by centripetal force. The force - acting in towars the centre - require to keep an object moving in a circle is calle centripetal force. (ii) State Newton s law of universal gravitation. Newton s law of gravitation states that any two point masses in the universe attract each other with a force that is irectly proportional to the prouct of their masses an inversely proportional to the square of the istance between them. (iii)derive the relationship between the perio of a satellite, the raius of its orbit an the mass of the Earth. Centripetal force = gravitational force mv /r = Gm1m/r v = Gm/r Also T = πr/v T = 4π r 3 /GM (iv) Calculate the height of a GPS satellite above the Earth s surface. T = 4π r 3 /GM r = m h = ( raius of the earth) = m (v) Calculate the spee of a GPS satellite. v = GM/r v = m s 1 (vi) Calculate the minimum time it takes a GPS signal to travel from the satellite to a receiver on the surface of the Earth. v = s/t t = s (vii) Explain why GPS satellites are not classe as geostationary satellites. Perioic time is not 4 hours // Not in the same place above the Earth (viii) Raio-waves, such as those use by GPS satellites, have the lowest frequency of all electromagnetic raiation types. What type of electromagnetic raiation has the next lowest frequency? Microwaves 014 Question 1 (a) (i) State Hooke s law. Hooke s Law states that when an object is stretche the restoring force is irectly proportional to the isplacement, provie the elastic limit is not exceee. (ii) What is the new length of the spring? F = k (extension). extension = F/k = ( )/1 = m New length = 41.3 mm (iii)sketch a velocity-time graph of the motion of the object. axes labelle perioic graph correct sinusoial shape, beginning with v = 0 (iv) Calculate the perio of oscillation of the object. ω = k/m (= 600) T = π/ω T = 0.56 s 1

2 013 Question 11 (a) Seismic waves can be longituinal or transverse. What is the main ifference between them? A Longituinal Wave is a wave where the irection of vibration is parallel to the irection in which the wave travels. A Transverse wave is a wave where the irection of vibration is perpenicular to the irection in which the wave travels. (b) How far is the station from the centre of the earthquake? s = vt = = m = 135 km (c) Draw a iagram to show the forces acting on the suspene mass when the seismometer is at rest. Weight acting ownwars an tension acting upwars () At rest, the tension in the spring is 49 N. What is the value, in kilograms, of the suspene mass? W = mg m = 5 kg (e) What type of motion oes the frame have when it moves relative to the mass? Simple Harmonic Motion (f) Give an equation for the acceleration of the groun in terms of the perioic time of the wave motion an the isplacement of the groun. a = s T = π ω a = 4π s T (g) If the perio of the groun motion was recore as 17 secons an its amplitue was recore as 0.8 cm, calculate the maximum groun acceleration at the recoring station. a max = 4π (0.008) 17 amax = m s - (h) Explain why an emf is generate in the coil. Magnetic fiel passing through the coil is changing OR The coil cuts the magnetic flux 013 Question 6 (i) State Newton s law of universal gravitation. Newton s Law of Gravitation states that any two point masses in the universe attract each other with a force that is irectly proportional to the prouct of their masses, an inversely proportional to the square of the istance between them. (ii) Explain what is meant by angular velocity. Angular Velocity is the rate of change of angle with respect to time. (iii)derive an equation for the angular velocity of an object in terms of its linear velocity when the object moves in a circle. θ θ = s r t = s tr But = /t an v = s/t: = v r Cross-multiply to get: v = r (iv) Calculate (a) the angular velocity, (b) the linear velocity, of the ISS. ω = π T = π 5570 = s -1 v =rω v = ( ) ( ) = m s 1 (v) Name the type of acceleration that the ISS experiences as it travels in a circular orbit aroun the earth. Centripetal / Gravitational??

3 (vi) What force provies this acceleration? Gravitational (vii) Calculate the attractive force between the earth an the ISS. mv F c r F = N (viii) Hence or otherwise, calculate the mass of the earth. F = GM 1M M 1 = F GM M1 = kg (other methos also acceptable can you think of two others?) (ix) Why o occupants of the ISS experience apparent weightlessness? They are in freefall // ISS accelerating at the same rate as occupants What is the perio of a geostationary communications satellite? One ay 01 Question 1 (a) (i) What is the angular velocity of the hammer uring its final swing? T = π = π 0.8 = 7.85 ra s -1 (ii) Even though the hammer moves at a constant spee, it accelerates. Explain. The irection changes (continuously) (iii)calculate the acceleration of the hammer uring its final swing. a = ω r a = (7.85) () a = m s - towars the centre of orbit (iv) Calculate the kinetic energy of the hammer as it is release. K.E. = ½ mv K.E. = ½ m(r ω) K.E. = 896 J 011 Question 6 (c) (i) Draw a iagram showing the forces acting on the chil as the merry-go-roun rotates. (ii) What is the maximum angular velocity of the merry-go-roun so that the chil will not fall from it, as it rotates? F = mω r 50 = 30 ω (.) ω = 0.84 ra s -1 (iii)if there was no force of friction between the chil an the merry-go-roun, in what irection woul the chil move as the merry-go-roun starts to rotate? The chil woul remain stationary / any appropriate answer. 3

4 011 Question 1 (a) (i) State Hooke s law. For a stretche string the restoring force is proportional to isplacement (ii) Why oes the boy vibrate with simple harmonic motion? The acceleration is proportional to the isplacement (iii)calculate the frequency of vibration of the boy? ω = 16 ω = 4 f = 1/T an T = π/ω so f = ω/π f = 0.64 Hz (iv) What is the magnitue of (i) the maximum force, (ii) the minimum force, which causes the boy s motion? a max = ( )16(0.05) = 0.80 (Fmax occurs when acceleration / isplacement is a maximum) Fmax = (0.50)(0.80) = 0.0 N Fmin = Question 6 (i) State Newton s law of universal gravitation. Force between any two point masses is proportional to prouct of masses an inversely/inirectly proportional to square of the istance between them. (ii) Use this law to calculate the acceleration ue to gravity at a height above the surface of the earth, which is twice the raius of the earth. Note that above surface is 3 from earth s centre GM g GM g new ( 3) where = m gnew = 1.09 m s - (iii)explain why the spacecraft continues on its journey to the moon, even though the engines are turne off. There are no external forces acting on the spacecraft so from Newton s 1 st law of motion the object will maintain its velocity. (iv) Describe the variation in the weight of the astronauts as they travel to the moon. Weight ecreases as the astronaut moves away from the earth an gains (a lesser than normal) weight as she/he approaches the moon (v) At what height above the earth s surface will the astronauts experience weightlessness? Gravitational pull of earth = gravitational pull of moon Gm E m Gm m m M M 1 E M 9 = ( 81) 1 1 E = 9 m an E + m = m 10 m =

5 m = E = Height above the earth = ( ) ( ) = m (vi) The moon orbits the earth every 7.3 ays. What is its velocity, expresse in metres per secon? v = πr T π( ) v = v = 10.9 m s -1 (vii) Why is there no atmosphere on the moon? The gravitational force is too weak to sustain an atmosphere. 009 Question 1 (a) (i) State Hooke s law. When a string is stretche the restoring force is proportional to the isplacement. (ii) When a sphere of mass 500 g is attache to a spring of length 300 mm, the length of the spring increases to 330 mm. Calculate the spring constant. When the mass of 500 g is attache the new force own = mg = (0.5)(g). Because the spring is in equiblibrium this must be equal to the force up (which is the restoring force). Hooke s law in symbols: F = k x g) = kx -1 (iii)the sphere is then pulle own until the spring s length has increase to 350 mm an is then release. Describe the motion of the sphere when it is release. It executes simple harmonic motion because the isplacement is proportional to t he acceleration. (iv) What is the maximum acceleration of the sphere? F = ma = kx a = kx/m = (163.3)(0.0)/(0.5) = 6.53 m s - OR a = = k/m = 163.3/0.5 a = 6.53 m s Question 6 (i) State Newton s law of universal gravitation. Newton s Law of Gravitation states that any two point masses in the universe attract each other with a force that is irectly proportional to the prouct of their masses, an inversely proportional to the square of the istance between them. (ii) What type of force is require to keep the ISS in orbit? Gravity (iii)what is the irection of this force? Towars the centre of the orbit / inwars / towars the earth (iv) Calculate the acceleration ue to gravity at a point 400 km above the surface of the earth. Gm m GM 1 = mg g - 11 )( ) / ( ) (v) An astronaut in the ISS appears weightless. Explain why. He is in a state of free-fall (the force of gravity cannot be felt). (vi) Derive the relationship between the perio of the ISS, the raius of its orbit an the mass of the earth. See notes Circular Motion chapter for a erivation. 5

6 (vii) Calculate the perio of an orbit of the ISS. = s (viii) After an orbit, the ISS will be above a ifferent point on the earth s surface. Explain why. The ISS has a ifferent perio to that of the earth s rotation (it is not in geostationary orbit). (ix) How many times oes an astronaut on the ISS see the sun rise in a 4 hour perio? ( ) = 16 ( sunrises). 007 Question 6 (i) State Hooke s law. For a stretche string the restoring force is proportional to the extension. (ii) A stretche spring obeys Hooke s law. When a small sphere of mass 300 g is attache to a spring of length 00 mm, its length increases to 85 mm. Calculate its spring constant. F = mg = ks (0.30)(9.8) = (k)(0.085) k = 34.6 N m -1 (iii)the sphere is pulle own until the length of the spring is 310 mm. The sphere is then release an oscillates about a fixe point. Derive the relationship between the acceleration of the sphere an its isplacement from the fixe point. F = - ks ma = - ks a = - (k/m)s a α -s a = - k s (iv) Why oes the sphere oscillate with simple harmonic motion? Its acceleration is proportional to its isplacement from a fixe point. (v) Calculate the perio of oscillation of the sphere. From above:ω = k/m ω = 34.6 / 0.3 T = 0.6 s (vi) Calculate the maximum acceleration of the sphere. This occurs when s is a maximum, i.e. when s = amplitue = = 0.05 m. a = -ω s a = - (10.7) (0.05) a = (-).89 m s - (vii) Calculate the length of the spring when the acceleration of the sphere is zero. This occurs at the fixe point when l = 0.85 m 006 Question 6 (i) Define velocity. Velocity is the rate of change of isplacement with respect to time. (ii) Define angular velocity. Angular velocity is the rate of change of angle with respect to time. (iii)derive the relationship between the velocity of a particle travelling in uniform circular motion an its angular velocity. θ = s /r θ /t = s/rt ω = v /r v = ω r (iv) A stuent swings a ball in a circle of raius 70 cm in the vertical plane as shown. The angular velocity of the ball is 10 ra s 1. What is the velocity of the ball? v = ω r = (10)(0.70) = 7.0 m s -1 (v) How long oes the ball take to complete one revolution? T= πr/v = π(0.70)/v = 0.63 s 6

7 (vi) Draw a iagram to show the forces acting on the ball when it is at position A. Weight (W) ownwars; reaction (R) upwars; force to left (ue to friction or curle fingers) (vii) The stuent releases the ball when is it at A, which is 130 cm above the groun, an the ball travels vertically upwars. Calculate the maximum height, above the groun, the ball will reach. v = u + as 0 = (7) + (-9.8) s / s =.5(0) m (viii) Calculate the time taken for the ball to hit the groun after its release from A. s = ut + ½ at = 7t ½ (9.8)t t = 1.59 s 005 Question 6 (i) Define angular velocity. Angular velocity is the rate of change of isplacement with respect to time. (ii) Define centripetal force. The force - acting in towars the centre - require to keep an object moving in a circle is calle Centripetal Force. (iii)state Newton s Universal Law of Gravitation. Newton s Law of Gravitation states that any two point masses in the universe attract each other with a force that is irectly proportional to the prouct of their masses, an inversely proportional to the square of the istance between them. (iv) Derive the relationship between the perio of the satellite, the mass of Saturn an the raius of the orbit. See notes on the Circular Motion chapter for the erivation. (v) The perio of the satellite is 380 hours. Calculate the raius of the satellite s orbit aroun Saturn. T = = s r 3 = T GM/4π r 3 = ( ) ( )( )/ 4π 9 m (vi) The satellite transmits raio signals to earth. At a particular time the satellite is m from earth. How long oes it take the signal to travel to earth? v = s/t ( ) = ( )/t t = 4000 s (vii) It is notice that the frequency of the receive raio signal changes as the satellite orbits Saturn. Explain why. Doppler Effect ue to relative motion between source of signal an the etector 7

8 004 Question 1 (a) (i) State Newton s universal law of gravitation. Newton s Law of Gravitation states that any two point masses in the universe attract each other with a force that is irectly proportional to the prouct of their masses, an inversely proportional to the square of the istance between them. (ii) Centripetal force is require to keep the earth moving aroun the sun. What provies this centripetal force? Gravitational pull of the sun. (iii)in what irection oes this centripetal force act? Towars the centre. (iv) Give an expression for centripetal force. mv r F c (v) Calculate the mass of the sun. Gm1m mv GM F g F c an r Equating gives R s = ( ) ( )/ = kg. v s = v R/G 00 Question 6 (i) State Newton s secon law of motion. Newton s Secon Law of Motion states that the rate of change of an object s momentum is irectly proportional to the force which cause it, an takes place in the irection of the force. (ii) Name this law an give a statement of it. Hooke s Law states that when an object is stretche the restoring force is irectly proportional to the isplacement, provie the elastic limit is not exceee. (iii)give the name for this type of motion an escribe the motion. Simple harmonic motion; an object is sai to be moving with Simple Harmonic Motion if its acceleration is irectly proportional to its istance from a fixe point in its path, an its acceleration is irecte towars that point. (iv) Give two other examples of systems that obey this law. Stretche elastic, penulum, oscillating magnet, springs of car, vibrating tuning fork, object bobbing in water waves, ball in saucer, etc. (v) Using the formula F = ks, calculate the value of k, the constant for the springs of the bike. 5 F = ks ks -k (.005) -k (.005) N m -1 (vi) Calculate the perio of oscillation of the cyclist. k/m = ω ω = 38 s -1 T = π/ω = 0.16 s (vii) Calculate the number of oscillations of the cyclist per secon. f = 1/T approximately = 6 8