Simple Harmonic Motion of Spring
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1 Nae P Physics Date iple Haronic Motion and prings Hooean pring W x U ( x iple Haronic Motion of pring. What are the two criteria for siple haronic otion? - Only restoring forces cause siple haronic otion. restoring force is a force that it proportional to the displaceent fro equilibriu and in the opposite direction. - Position, velocity and the other variables of siple haronic otion are sinusoidal functions of tie.. he diagra to the right shows a g bloc attached to a Hooean spring on a frictionless surface. he bloc experiences no net force when it is at position. When the bloc is to the left of point the spring pushes it to the right. When the bloc is to the right of point, the spring pulls it to the left. he ass is pulled to the right fro point to point and released at tie t =. he bloc then oscillates between positions and. ssue that the syste consists of the bloc and the spring and that no dissipative forces act. a) he bloc taes 4. s to ae oscillations. What is the period of oscillation for this syste? = 4s/ oscillations = sec/oscillation b) What is the frequency of this oscillating syste? f = / = ½ =.5 Hz c) What is the aplitude of vibration of this syste? =. d) Deterine the spring constant of the spring. ) x cos( t) cos( a e) Write an equation that describes the position of the ass as a function of tie, starting fro position at tie t =. x. cos(. cos( t) x 9.7N /
2 f) Explain what would happen to the period and frequency of this syste if you were to double the aplitude while eeping the ass and spring constant the sae. he period and frequency would not change since they do not depend on the aplitude of oscillation g) Explain what would happen to the period and frequency of this syste if you were to double the ass while eeping the aplitude and spring constant the sae. period depends on square root of ass so if ass doubled, would change by frequency depends on /( ) so if ass doubled, f would change by / =.7 x x.4 f.7f h) Explain what would happen to the period and frequency of this syste if you were to double the spring constant while eeping the aplitude and ass constant. period depends on /( ) so if spring constant doubled, would change by / =.7x frequency depends on square root of so if spring constant doubled, f would change by x f.4f x x i) Deterine the aount of energy of the oscillating spring and ass syste. he energy of the bloc and spring is conserved since only the spring force (a conservative force) does wor. herefore E = E = E E = E = ½ (x ) = ½ (9.7)( ) =.394J.7, j) Where does the bloc have axiu speed and what is the axiu speed? loc has axiu speed at point, the equilibriu point of the spring. t that point, all the energy is in the for of inetic energy E E V.394 v () v.63 ) Where does the bloc experience axiu acceleration and what is the axiu acceleration? loc has axiu acceleration at points and, at ax displaceent fro equilibriu, where net = s pring is greatest a x (9.7)(.).97 l) What would happen to the energy if the aplitude of oscillation were tripled while eeping the ass and spring constant unchanged? he energy is proportional to the aplitude squared (E = ½ ). herefore, if the aplitude of the oscillation were tripled, the total energy would increase 9xs. ) What would happen to the axiu speed if the aplitude of oscillation were tripled while eeping the ass and spring constant unchanged? he ax speed at equilib is proportional to the aplitude (E =E = ½ = ½ v ). herefore, if the aplitude of the oscillation were tripled, the ax speed would also triple.
3 3. 5 g ass is attached to a spring that is hanging vertically. he spring is stretched.5 fro its equilibriu position. a) What is the spring constant? x g g g x 96N / x=.5 b) What ass would be required to stretch the spring three ties the distance? g 3xs or 5 g since the ass is directly proportional to the displaceent he 5 g object hanging on the spring is allowed to coe to its new equilibriu. It is then set in otion by stretching it a further.3. he ass oscillates in siple haronic otion c) What is the period of the oscillation? 5. s g ass on a spring is stretched and released. he period of oscillation is easured to be.46 s. What is the spring constant? N / 5. weight in a spring-ass syste exhibits haronic otion. he syste is in equilibriu when the weight is otionless. If the weight is pulled down or pushed up and released, it would tend to oscillate freely if there were no friction. In a certain spring-ass syste, the weight is 5 feet below a -foot ceiling when it is at rest. he otion of the weight can be described by the equation, y = 3sin(t), where y is the distance fro the equilibriu point, and t is easured in seconds. a) ind the period of the otion. seconds. he expression in the sin function, t, is equal to ft. herefore f =.5s and = s b) What is the frequency of the otion? / cycle per second. c) ind the aplitude of the otion. 3ft d) How far fro the ceiling is the weight after 3.5 seconds? 8 ft sp
4 6. he graph in the figure shows the displaceent of a.4 g particle fro a fixed equilibriu position. Z Z V V Z Use the graph to deterine: a) the period of otion Period is tie of cycle =.s b) the axiu speed of the particle during oscillation, he ax speed is when the object passes through the equilibriu where there is no elastic potential energy. Rather all the energy is in the for of inetic energy. he energy of the particle and spring is conserved since only the spring force (a conservative force) does wor. We need to find the echanical energy before finding the ax speed. E = E = ½ = ½ (4)( ) = 8J E = E = ½v = ½(.4)v = 8J v = 63. /s.4. 4N / c) the axiu acceleration experienced by the particle. Particle has axiu acceleration at ax displaceent fro equilibriu, where net = s pring is greatest x (4)() a.4 ax ax / On the graph, ar the following a) a point where the velocity is zero ( label this as Z) b) a point where the velocity is positive and has the largest agnitude ( label this as V) c) a point where the acceleration is positive and has the largest agnitude ( label this as ). s
5 Kinetic E Potential E net (N) slope a (/s ) v (/s) slope x () Motion sensor 4.c 7. he diagra above shows a g bloc attached to a Hooean spring on a frictionless surface. he bloc experiences no net force when it is at position. When the bloc is to the left of point the spring pushes it to the right. When the bloc is to the right of point, the spring pulls it to the left. he ass is pulled to the left fro point to point and released. he bloc then oscillates between positions and. otion sensor placed to the right of position gathers position-tie data for the oscillating bloc. he position vs. tie graph below describes the otion of this syste for four cycles sec x(t)=.3sin(+.5 =.3sin(t)+.5 v(t)=.9cos( =.9cos(t) a(t)= -.8sin( = -.8sin(t) (t)= -3.6sin( = -3.6sin(t) J 3.5J U (t)= ½(x ) = 3.5sin (t) K(t)= ½v = 3.5cos (t)
6 a) What is the period of oscillation for this syste? = sec b) What is the frequency of this oscillating syste? f = / = Hz c) U (t)= ½(x ) = 3.5sin (t) d) What is the aplitude of vibration of this syste? =.3 e) Deterine the spring constant of the spring f) oplete setches for the other graphs shown based on the position vs. tie graph. Label axia and inia with nuerical values on the axes. oe things to consider when you setch the graphs: - You can ae qualitative v-t and a-t setches by considering the slopes of appropriate graphs o get a qualitative v-t graph (no nubers), tae the slope of the x-t graph at points where it is zero, ax or in and plot the v-t points shown. onnect points with a sinusoidal curve o get a qualitative a-t graph (no nubers), tae the slope of the v-t graph at points where it is zero, ax or in and plot the a-t points shown. onnect points with a sinusoidal curve -t graph loos qualitatively lie a-t since they are directly related by ultiplying by the scalar. Energy graphs are related to x (potential) and v (inetic) so the negative regions on the x-t or v-t becoe positive on the energy graphs. - Reeber that Us = ½ x where x = x-x is displaceent fro equilibriu (x = at the equilibriu position) Us is NO ½ x when x = at the otion sensor. - Reeber that a = net/ = (-x)/ how calculations necessary to label the graphs (ax and in values of each variable) Velocity Graph cceleration Graph v ax is at the equilibriu point where all the energy is inetic energy. he energy of the ass-spring syste is conserved at every position; E = E = E = E x v( t) vax cos( v E v v v E orce Graph ax 78.7(.3.88 net net a a ) ax (.8) 3.6N cos( 78.7N / a( t) aax sin( Max acceleration is at ax displaceent (x=). net x a a ax ax (78.7)(.3) a a a.8 ax
7 Potential Energy Graph U is proportional to x, not to x. x = x-x where x is the equilibriu position of the spring and bloc. ro the x(t) graph, you can see that x is.5. Zero potential energy can be set anywhere. or convenience, define it to be the equilibriu point of the spring and bloc. Max U is at ax x which is the aplitude (.3) ( ) 3.54J KineticEnergy Graph has sae ax as Potential E graph since energy is conserved g) What is the average velocity of the bloc during one cycle? v x U U ax ( x ) h) What is the average speed of the bloc during one cycle? In one cycle, the distance travelled can be seen on the x-t graph s d.. s / i) If the frequency of the oscillation were doubled, what would the average speed of the bloc be during one cycle? he average speed would double since the bloc would travel the sae distance in ½ the tie d. s.4.5
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