Tutorial 1 Simple Harmonic Motion

TUTORIAL 1 SIMPLE HARMONIC MOTION

Instructor: Kazumi Tolich About tutorials

¨ Tutorials are conceptual exercises that should be worked on in groups.

¨ Each slide will consist of a series of questions that you should discuss with the students sitting around you.

¨ There will be a few clicker questions per session. Clicker questions are shown in red.

¨ There will be several teaching assistants wandering around the room to assist you. If you are having difficulty, they will ask you leading questions to help you understand the idea. I. Hook’s law for spring SIMPLE HARMONIC MOTION Mech 3 79 ¨ A block of mass m on a frictionless surface is attached to an ideal spring, as shown in figure 1. I. TheHooke’s spring, law with for a springsspring constant k, is fixed to the wall. The dashed line indicates the position A blockof the of right mass edge m on of a frictionlessthe spring when surface the is spring attached is relaxed,to an ideal and massless the block spring, is at as its shown equilibrium in figureposition. 1. The Figures spring, 2 whandich 3 has show spring the blockconstant held k, inis placefixed toa thedistance wall. AThe to dashedthe right line and indicates left of the the position of the right edge of the spring when the spring is neither stretched nor compressed andequilibrium the block is position at its equilibrium, respectively position.. Block is at rest Block is at rest Block is at rest

m m m A A

Figure 1 Figure 2 Figure 3 Figures 2 and 3 show the block held in place a distance A to the right and left of the equilibrium position, respectively.

A. In the boxes at right, draw arrows to represent Arrows for figure 2 Arrows for figure 3 the directions of: Position of block Position of block • the position of the block, x, taking x = 0 when the block is at its equilibrium position and • the force on the block by the spring, F. Force on block Force on block Use the arrows that you drew to explain why the by spring by spring minus sign is necessary in the expression F = –kx (Hooke’s law for an ideal spring).

B. What is the net force on the block when it is held in place as shown in figure 2? Explain.

Suppose the hand in figure 2 were suddenly removed. After the hand is removed, how would the force on the block by the spring be related to the net force on the block? Explain.

C. Use Newton’s second law to write an expression for the acceleration, a, of the block in terms of x, k, and m for an instant after the block has been released.

Tutorials in Introductory Physics ©Pearson Custom Publishing McDermott, Shaffer, & P.E.G., U. Wash. Updated Preliminary Second Edition, 2011 SIMPLE HARMONIC MOTION Mech 79

I. Hooke’s law for springs A block of mass m on a frictionless surface is attached to an ideal massless spring, as shown in figure 1. The spring, which has spring constant k, is fixed to the wall. The dashed line indicates the position of the right edge of the spring when the spring is neither stretched nor compressed and the block is at its equilibrium position. Block is at rest Block is at rest Block is at rest

SIMPLE HARMONIC MOTION Mech 79

I. Hooke’s law for springs m m m I.A block ofHook’s mass m on a frictionless law surface is attached for to an ideal spring massless spring, as shown in figure 1. The spring, which has spring constant k, is fixed to the wall. The dashed line indicates A A the position of the right edge of the spring when the spring is neither stretched nor compressed 4 and the block is at its equilibrium position. Figure 1 Figure 2 Figure 3 Block is at rest Block is at rest Block is at rest Figures 2 and 3 show the block held in place a distance A to the right and left of the equilibrium position, respectively. m m m A A A. In the boxes at right, draw arrows to represent Arrows for figure 2 Arrows for figure 3 Figure 1 Figure 2 Figure 3 the directions of: Position of block Position of block Figures 2 and 3 show the block held in place a distance A to the right and left of the equilibrium A. position,In the respectively. boxes at right, draw arrows• the position to represent of the block, the directions x, taking x of: = 0 when the block is at its equilibrium position and ¤ The position of the block, �, taking � = 0 when the block is at the equilibrium A. In the boxes at right, draw arrows to represent Arrows for figure 2 Arrows for figure 3 position. • the force on the block by the spring, F. Force on block Force on block the directions of: Position of block Position of block ¤ The force on the block by the spring, �⃗. by spring by spring • the position of the block, x, taking x = Use0 when the arrows that you drew to explain why the ¤ theUse block the is arrowsat its equilibrium that you position drew andminus to explain sign iswhy necessary the minus in sign the expressionis necessary in F = –kx (Hooke’s law for an ideal spring). • theHook’s force on law: the block�⃗ = by− the�� .spring, F. Force on block Force on block by spring by spring Use the arrows that you drew to explain why the B. Whatminus sign is theis necessary net force in the expressionon the block when it is held in place as shown in F = –kx (Hooke’s law for an ideal spring). figure 2?

B. What is the net force on the block when it is held in place as shown in figure 2? Explain. B. What is the net force on the block when it is he ld in place as shown in figure 2? Explain.

Suppose the hand in figure 2 were suddenlySuppose removed. the After hand the hand in figureis removed, 2 were how suddenly removed. After the hand is removed, how would the force on the block by the springwould be related the to theforce net forceon the on theblock block? by the spring be related to the net force on the block? Explain. Explain.

C. Use Newton’s second law to write an expression for the acceleration, a, of the block in terms of x, k, and m for an instant after the block has been released. C. Use Newton’s second law to write an expression for the acceleration, a, of the block in terms of x, k, and m for an instant after the block has been released.

Tutorials in Introductory Physics ©Pearson Custom Publishing McDermott, Shaffer, & P.E.G., U. Wash. Updated Preliminary Second Edition, 2011

Tutorials in Introductory Physics ©Pearson Custom Publishing McDermott, Shaffer, & P.E.G., U. Wash. Updated Preliminary Second Edition, 2011 Quiz: T1-1 answer

¨ The net force is zero.

¨ The block is not accelerating, so according to Newton's 2nd law, the net force on it is zero. ¨ ∑ �⃗ = �� SIMPLE HARMONIC MOTION Mech 79

I. Hooke’s law for springs I.A block ofHook’s mass m on a frictionless law surface is attached for to an ideal spring massless spring, as shown in figure 1. The spring, which has spring constant k, is fixed to the wall. The dashed line indicates the position of the right edge of the spring when the spring is neither stretched nor compressed 6 and the block is at its equilibrium position. Block is at rest Block is at rest Block is at rest

m m m A A

Figure 1 Figure 2 Figure 3 Figures 2 and 3 show the block held in place a distance A to the right and left of the equilibrium position,¤ Suppose respectively. the hand in figure 2 were suddenly removed. After the hand is

A. In theremoved, boxes at right, draw how arrows towould represent theArrows force for figure on 2 Arrowsthe forblock figure 3 by the spring be related to the the directions of: Position of block Position of block • thenet position force of the block, on x , thetaking xblock = 0 when ? the block is at its equilibrium position and C. • theUse force onNewton’s the block by the spring, second F. lawForce on toblock writeForce onan block expression for the Use the arrows that you drew to explain why the by spring by spring minus sign is necessary in the expression F =acceleration, –kx (Hooke’s law for an ideal �spring)., of the block in terms of �, �, and � for an instant

after the block has been released.

B. What is the net force on the block when it is held in place as shown in figure 2? Explain.

Suppose the hand in figure 2 were suddenly removed. After the hand is removed, how would the force on the block by the spring be related to the net force on the block? Explain.

C. Use Newton’s second law to write an expression for the acceleration, a, of the block in terms of x, k, and m for an instant after the block has been released.

Tutorials in Introductory Physics ©Pearson Custom Publishing McDermott, Shaffer, & P.E.G., U. Wash. Updated Preliminary Second Edition, 2011 Mech Simple harmonic motion II.80 Simple harmonic motion II. Simple harmonicMech Simple motion harmonic motion 7 80 A pendulum is hung directly above the block-spring system ¨ A pendulum with a massII. � Simpleis hung harmonic directly motion above the block- from section I. The blockA pendulum and is the hung pendulum directly above thebob block-spring both have system springmass system m. The from spring section constantfrom I.section The isI. length Thek, andblock oftheand the length pendulum pendulum of bob the both is have �. mass m. The spring constant is k, and the length of the Thependulum parameters is l. of The the parameters systemspendulum is havel. Theof theparameters been systems chosen of the systemshave such havebeen that been l l chosen such that the blockchosen such is always that the block directly is always belowdirectly below the the the block is always directlypendulum below bob. the pendulum bob. pendulum bob. A. Draw a vector to representA. In the spacesthe netbelow, force draw a onvector the to represent block the and net force on each object when it is on the far left as shown. A. In the spaces below, draw a vector to represent the net m the bob when it is on the farNet force left on blockas shown.Net force on bob force on each object when it is on the far left as shown. k m Net force on block Net force on bob m Equilibriumk position m

B. Suppose that the masses of the block and the pendulum bob were doubled, and the block and bob were then released from rest at the far left position. Equilibrium position The following questions serve as a guide to help you determine whether the block would remain directly below the pendulum bob at all times. 1. As a result of doubling the masses of the block and pendulum bob:

Would the magnitude of the net force on each object when it is at the far left B. Suppose that the massesincrease of the by a block factor of and2, decrease the bypendulum a factor of 2, bobor remain were the same? doubled, Explain. and the block and

bob were then released• from Fnet, block rest at the far left position.

The following questions serve as a guide to help you determine whether the block would • F remain directly below thenet, pendulum bob bob at all times.

Would the magnitude of the acceleration of each object when it is at the far left 1. As a result of doublingincrease the by a massesfactor of 2, ofdecrease the byblock a factor and of 2, pendulumor remain the same? bob: Explain.

Would the magnitude• ablock of the net force on each object when it is at the far left increase by a factor of 2, decrease by a factor of 2, or remain the same? Explain. • abob

• Fnet, block Would the time it takes each object to travel from the far left position to the equilibrium position increase, decrease, or remain the same? Explain.

• Δtblock • Fnet, bob

• Δtbob Would the magnitude of the acceleration of each object when it is at the far left increase by a factor of 2, decrease by a factor of 2, or remain the same? Explain.

• ablock

• abob

Would the time it takes each object to travel from the far left position to the equilibrium position increase, decrease, or remain the same? Explain.

• Δtblock

• Δtbob

Quiz: T1-2 answer Mech Simple harmonic motion 8 80

¨ The net force on the bob isII. tangent Simple harmonic to its path motion A pendulum is hung directly above the block-spring system toward the equilibrium positionfrom section. I. The block and the pendulum bob both have mass m. The spring constant is k, and the length of� the ¨ The bob is not acceleratingpendulum radially is l. from The parameters or of the systems have been l toward the pivot. The directionchosen ofsuch acceleration that the block is always directly below the pendulum bob. does not have any component in the radial A. In the spaces below, draw a vector to represent the net direction. The component of theforce weight on each objectin the when it is on the far left as shown. m radial direction is balanced withNet the force tension on block in Net force on bob the string. k m ¨ The weight of the bob has a component tangent to the path toward the equilibrium Equilibrium position position. �

B. Suppose that the masses of the block and the pendulum bob were doubled, and the block and bob were then released from rest at the far left position. The following questions serve as a guide to help you determine whether the block would remain directly below the pendulum bob at all times. 1. As a result of doubling the masses of the block and pendulum bob: Would the magnitude of the net force on each object when it is at the far left increase by a factor of 2, decrease by a factor of 2, or remain the same? Explain.

• Fnet, block

• Fnet, bob

Would the magnitude of the acceleration of each object when it is at the far left increase by a factor of 2, decrease by a factor of 2, or remain the same? Explain.

• ablock

• abob

Would the time it takes each object to travel from the far left position to the equilibrium position increase, decrease, or remain the same? Explain.

• Δtblock

• Δtbob

II. Simple harmonic motion

¨ Suppose that the masses of the block and the pendulum bob were doubled, and the block and bob were then released from rest at the far left position.

¨ The following questions serve as a guide to help you determine whether the block would remain directly below the pendulum bob at all times.

1. As a result of doubling the masses of the block and pendulum bob, would the following quantities increase by a factor of 2, decrease by a factor of 2, or remain the same?

¤ the magnitude of the net force on the block, �, , or the bob, �, , when it is at the far left Quiz: T1-3 answer

¨ increase by a factor of 2

¨ The weight of the bob doubles, so does its component � tangent to the path.

� II. Simple harmonic motion

1. As a result of doubling the masses of the block and pendulum bob,

¤ would the magnitude of the acceleration of each object, � or �, when it is at the far left increase by a factor of 2, decrease by a factor of 2, or remain the same?

¤ would the time it takes for each object to travel from the far left position to the equilibrium position, ∆� or ∆�, increase, decrease, or remain the same?

2. Check that your answers regarding ∆� and ∆� are consistent with the relationships below: (Hint: For each object, what is the relationship between ∆� and the period of oscillation, �?) ¤ � = 2� �⁄� (period of oscillation for a mass on a spring) ¤ � = 2� �⁄� (period of oscillation for a simple pendulum)

3. Will the block and pendulum bob still move together after their masses have been doubled? If not, describe what additional changes could be made so that the block and pendulum bob would again move together. Quiz: T1-4 answer

¨ No ¨ The period of oscillation is increased by 2 for the block, but it remains the same for the bob. ¨ ∆� = � = 2� �⁄� = � �⁄� (depends on mass) ¨ ∆� = � = 2� �⁄� = � �⁄� (independent of mass) Simple harmonic motion Mech 81

2. Check that your answers regarding Δtblock and Δtbob are consistent with the relationships below: (Hint: For each object, what is the relationship between Δt and the period of oscillation, T?)

• Tblock = 2π m/k (period of oscillation for a mass on a spring)

• Tbob = 2π l/g (period of oscillation for a simple pendulum)

3. Will the block and pendulum bob still move together after their masses have been III. Energy ofdoubled? a simple If not, describe whatharmonic additional changes couldoscillator be made so that the block and pendulum bob would again move together. 13

¨ The diagram at right is a plot of the total III. Energy of a simple harmonic oscillator E energy of a Thehorizontal diagram at right block is a plot- springof the total systemenergy of a as E a function ofhorizontal the position block-spring of system the as block a function with of the tot position of the block with respect to its equilibrium respect to itsposition. equilibrium The block oscillatesposition. with a Themaximum block distance from equilibrium of A. oscillates with a maximum distance from A. What feature of the diagram shows that the total equilibrium of Aenergy. of the system is conserved as the block oscillates?

-A +A

B. Determine what fraction of the total energy is potential energy when the block is at (1) x = +A and (2) x = +A/2. (Hint: U = 1/2 kx2.)

On the same axes above, plot the potential energy stored in the spring, U, as a function of x. Your graph should correctly reflect your answers above. C. On the same axes, plot the kinetic energy of the block, K, as a function of x. (Hint: What function must be added to the potential energy to equal the total energy of the system?) Label your graphs so that you can easily distinguish the kinetic energy K from the potential energy U. D. Is the time that the block takes to move from x = 0 to x = +A/2 greater than, less than, or equal to the time that the block takes to move from x = +A/2 to x = +A? Explain how your answer is consistent with your graph of the kinetic energy.

Tutorials in Introductory Physics ©Pearson Custom Publishing McDermott, Shaffer, & P.E.G., U. Wash. Updated Preliminary Second Edition, 2011 Simple harmonic motion Mech 81

2. Check that your answers regarding Δtblock and Δtbob are consistent with the relationships below: (Hint: For each object, what is the relationship between Δt and the period of oscillation, T?)

• Tblock = 2π m/k (period of oscillation for a mass on a spring)

III. Energy of a simple• Tbob =harmonic 2π l/g (period of oscillation foroscillator a simple pendulum)

14 3. Will the block and pendulum bob still move together after their masses have been doubled? If not, describe what additional changes could be made so that the block and A. What feature of the diagram shows thatpendulum the total bob would energy again ofmove the together. system is conserved as the block oscillates ?

B. Determine what fraction of the total energy is potential energy, � = 1⁄2 �� =III.+ Energy� of a� simple= + harmonic�⁄2 oscillator E , when the block is at The diagramand at right is a plot of the. P totallot theenergy of a E potential energy stored in the springhorizontalas a block-spring function systemof �. as a function of the tot position of the block with respect to its equilibrium C. On the same axes, plot the kinetic position.energy The of block the oscillates block, with �, aas maximum a function distance from equilibrium of A. of �. (Hint: What function must be added to the potential energy to A. What feature of the diagram shows that the total equal the total energy of the system?)energy Label of the your system graphs is conserved so asthat the block you can easily distinguish the kinetic energyoscillates? � from the potential energy �.

x D. Is the time that the block takes to move from � = 0 to � = + �⁄2 -A +A greater than, less than, or equal to the time that the block takes to move

from � = + �⁄2 to � = +�? ExplainB. Determine how your what fractionanswer of theis totalconsistent energy is potential energy when the block is at 2 with your graph of the kinetic energy.(1) x = +A and (2) x = +A/2. (Hint: U = 1/2 kx .)

Tutorials in Introductory Physics ©Pearson Custom Publishing McDermott, Shaffer, & P.E.G., U. Wash. Updated Preliminary Second Edition, 2011 Quiz: T1-5 answer

¨ Less than

¨ As the block moves from � = 0 to � = + �⁄2, the kinetic energy is always greater than its kinetic energy as the block moves from � = + �⁄2 to � = +�.

¨ So, the speed of the block is also always greater than its as the block moves from � = + �⁄2 to � = +�.

¨ The faster it moves, the less time it takes to cover the same distance. Simple harmonic motion Mech 81

2. Check that your answers regarding Δtblock and Δtbob are consistent with the relationships below: (Hint: For each object, what is the relationship between Δt and the period of oscillation, T?)

• Tblock = 2π m/k (period of oscillation for a mass on a spring)

III. Energy of a simple• Tbob =harmonic 2π l/g (period of oscillation foroscillator a simple pendulum)

16 3. Will the block and pendulum bob still move together after their masses have been doubled? If not, describe what additional changes could be made so that the block and E. Consider two points, P and Q, to the rightpendulum of bob the would block’s again move together. equilibrium position. Point P has position � = + �⁄2. Point Q is

the point for which the kinetic energy and the potential energy of the system are each equal to halfIII. Energythe total of a simple energy. harmonic oscillator E The diagram at right is a plot of the total energy of a Etot 1. Is point Q to the left of, to the righthorizontal of, or block-spring at the same system position as a function as point of the position of the block with respect to its equilibrium P? Mark the locations of points Pposition. and Q The on block the oscillates�-axis above.with a maximum distance from equilibrium of A. 2. When the block is at point P, is the kinetic energy of the system greater A. What feature of the diagram shows that the total than, less than, or equal to the potentialenergy energy? of the system Explain is conserved how as you the blockcan tell from the graph. oscillates?

3. Calculate the ratio of the kinetic energy to the potential energy when x

-A +A the block is at point P (i.e., at � = + �⁄2).

B. Determine what fraction of the total energy is potential energy when the block is at (1) x = +A and (2) x = +A/2. (Hint: U = 1/2 kx2.)

Tutorials in Introductory Physics ©Pearson Custom Publishing McDermott, Shaffer, & P.E.G., U. Wash. Updated Preliminary Second Edition, 2011 Quiz: T1-6

¨ 3 ¨ The total mechanical energy is constant and given by � = � + � = �� . ¨ This can be obtained at the turning point, where � = 0, and all the mechanical energy is due to the potential energy. ¨ Potential energy at � = + �⁄2 is � = � + = �� . ¨ Kinetic energy at � = + �⁄2 is � = � − � = �� − �� = �� . ¨ The ratio is = = 3 Simple harmonic motion Mech 81

2. Check that your answers regarding Δtblock and Δtbob are consistent with the relationships below: (Hint: For each object, what is the relationship between Δt and the period of oscillation, T?)

• Tblock = 2π m/k (period of oscillation for a mass on a spring)

III. Energy of a simple• Tbob =harmonic 2π l/g (period of oscillation foroscillator a simple pendulum)

18 3. Will the block and pendulum bob still move together after their masses have been doubled? If not, describe what additional changes could be made so that the block and pendulum bob would again move together. F. Suppose that the block were replaced by a

block with half the mass and released from rest

III. Energy of a simple harmonic oscillator E at � = +�. The diagram at right is a plot of the total energy of a E horizontal block-spring system as a function of the tot 1. Describe any resulting changesposition of the blockin the with respectthree to its energy equilibrium position. The block oscillates with a maximum distance graphs (�, �, �). Explain.from equilibrium of A. A. What feature of the diagram shows that the total 2. Is it possible to determineenergy the ofperiod the system is of conserved oscillation as the block oscillates? of a mass-spring system using information from x

-A +A energy graphs alone? If so, describe the steps you

would take to determineB. theDetermine period. what fraction If ofnot, the total state energy is potential energy when the block is at what other information you(1) xwould = +A and (2) need. x = +A/2. (Hint: U = 1/2 kx2.)

Tutorials in Introductory Physics ©Pearson Custom Publishing McDermott, Shaffer, & P.E.G., U. Wash. Updated Preliminary Second Edition, 2011 Quiz: T1-7 answer

¨ No

¨ From the amplitude and the total mechanical energy, you can determine the spring constant since at the turning point, � = � + � = �� + 0 = ��. ¨ But to determine the oscillation period, you need to know the mass: � = 2� �⁄�. III. Energy of a simple harmonic oscillator

20 Simple harmonic motion Mech G. Suppose instead that the original block were released from rest 83

at � = − �⁄2 and movedG. between Suppose instead the positions that the original � = block− �were⁄2 releasedand from rest at x = -A/2 and moved � = + �⁄2. between the positions x = -A/2 and x = +A/2. 1. As a result of this change, would1. Asthe a totalresult ofenergy this change,increase would, decrease,the total energy or E remain the same? Explain. increase, decrease, or remain the same? Explain. Etot, old 2. On the axes at right, graph the total energy, �,, potential energy � , and kinetic energy � for the new motion. 3. Does any part of the new set of graphs exactly coincide with any part

of the previous set of graphs? Explain.

4. Is the time that the block takes to move from � = 0 to � = + �⁄2 greater than, less than, or equal2. toOn the the timeaxes atthat right, the graph block the totaltook energy to move E , potential energy U , and kinetic energy x ⁄ total, new new -A +A from � = 0 to � = + � 2 beforeK thenew for amplitude the new motion. of oscillation was reduced? Explain. 3. Does any part of the new set of graphs exactly coincide with any part of the previous set of graphs? Explain.

4. Is the time that the block takes to move from x = 0 to x = +A/2 greater than, less than, or equal to the time that the block took to move from x = 0 to x = +A/2 before the amplitude of oscillation was reduced? Explain.

Check your answers with a tutorial instructor.

Tutorials in Introductory Physics ©Pearson Custom Publishing McDermott, Shaffer, & P.E.G., U. Wash. Updated Preliminary Second Edition, 2011 Quiz: T1-8 answer

¨ Greater than ¨ The total mechanical energy is reduced to � = � + = � . ¨ The potential energy curve between � = − �⁄2 and � = + �⁄2 remains the same.

¨ So, the kinetic energy between � = − �⁄2 and � = + �⁄2 is always smaller in the new case than the old case, implying that the block is moving slower in the new case.