Practice: A mass-spring system with an angular frequency ω = 8π rad/s oscillates back and forth. (a) Assuming it starts from rest, how much time passes before the mass has a speed of 0 again? (b) How many full cycles does the system complete in 60s?

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Spring Force (Hooke's Law) | 15 mins | 0 completed | Learn |

Intro to Simple Harmonic Motion (Horizontal Springs) | 32 mins | 0 completed | Learn |

Energy in Simple Harmonic Motion | 22 mins | 0 completed | Learn |

Simple Harmonic Motion of Vertical Springs | 20 mins | 0 completed | Learn |

Simple Harmonic Motion of Pendulums | 32 mins | 0 completed | Learn |

Energy in Pendulums | 16 mins | 0 completed | Learn |

Practice: A mass-spring system with an angular frequency ω = 8π rad/s oscillates back and forth. (a) Assuming it starts from rest, how much time passes before the mass has a speed of 0 again? (b) How many full cycles does the system complete in 60s?

Practice: A 4-kg mass on a spring is released 5 m away from equilibrium position and takes 1.5 s to reach its equilibrium position. (a) Find the spring’s force constant. (b) Find the object’s max speed.

Practice: What is the equation for the position of a mass moving on the end of a spring which is stretched 8.8cm from equilibrium and then released from rest, and whose period is 0.66s? What will be the object’s position after 1.4s?

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Concept #1: Intro to Simple Harmonic Motion

Example #1: Example

Practice #1: Practice

Concept #2: Equations of Simple Harmonic Motion

Practice #2: Practice

Example #2: Example

Practice #3: Practice

Example #3: Example

A spring with spring constant k oscillates with a total mass m. The shortest time interval between the instant of greatest speed and the instant when it turns around is

Objective: To understand how the two standard ways to write the general solution to a harmonic oscillator are related.There are two common forms for the general solution for the position of a harmonic oscillator as a function of time t:x(t)=A cos(ωt+ϕ) andx(t)=C cos(ωt)+S sin(ωt).Either of these equations is a general solution of a second-order differential equation (F⃗ =ma⃗ ); hence both must have at least two--arbitrary constants--parameters that can be adjusted to fit the solution to the particular motion at hand. (Some texts refer to these arbitrary constants as boundary values.)A)Find analytic expressions for the arbitrary constants C and S in Equation 2 (found in Part B) in terms of the constants A and ϕ in Equation 1 (found in Part A), which are now considered as given parameters.Give your answers for the coefficients of cos(ωt) and sin(ωt), separated by a comma. Express your answers in terms of A and ϕ.B)Find analytic expressions for the arbitrary constants A and ϕ in Equation 1 (found in Part A) in terms of the constants C and S in Equation 2 (found in Part B), which are now considered as given parameters.Express the amplitude A and phase ϕ (separated by a comma) in terms of C and S.

True or false. Increasing the mass attached to a spring will decrease the period of its vibrations.

The figure below shows two examples of SHM (simple harmonic motion), labeled A and B. For each, what is (a) the amplitude, (b) the period, and (c) the frequency and the angular frequency? (d) Write the equations for both A and B in the form of a sine or cosine.

The displacement of an oscillating object as a function of time is shown in the figure.a)What is the frequency?b)What is the amplitude?c)What is the period?d)What is the angular frequency of this motion?

1. For a transverse sinusoidal wave in a string, each point on the string is:a) moving opposite to the direction of the waveb) moving in the same direction as the wavec) undergoing simple harmonic motiond) stationary2. A wave has a frequency of 1 Hz and a wavelength of 2 m. What is the wave velocity?a) 2 m/sb) 1 m/sc) Not enough informationd) 0.5 m/s

1. What is the amplitude of the oscillation shown in (Figure 1) ?Express your answer using two significant figures2. What is the frequency of the oscillation shown in the figure?Express your answer using two significant figures.

An oscillator creates periodic waves on a stretched string.1. If the period of the oscillator doubles, what happens to the wavelength and wave speed?a) The wavelength doubles but the wave speed is unchanged.b) The wavelength is halved but the wave speed is unchanged.c) The wavelength is unchanged but the wave speed doubles.2. If the amplitude of the oscillator doubles, what happens to the wavelength and wave speed?a) Both wavelength and wave speed are unchanged.b) The wavelength is unchanged but the wave speed doubles.c) The wavelength doubles but the wave speed is unchanged.

Suppose you have a mass m attached to a spring constant k (N/m). The mass rests on a horizontal frictionless surface. Its equilibrium position is at x= 0. It is pulled aside a distance A and released.a) What is its speed as it passes the position x=0?b) What is the net force on the mass at position x=A?c) Do you expect the mass to have SHM?d) What is its speed when x=A/4?

What term denotes the time for one cycle of a periodic process?1-amplitude2-period3-frequency4-wavelengthFor vibrational motion, what term denotes the maximum displacement from the equilibrium position?1-amplitude2-period3-wavelength4-frequency

The diagram shows the motion of a simple harmonic oscillator.a. What is the amplitude of the motion (in cm)?b. What is the period (in s)?c. What is the equilibrium position (in cm)?d. What is the maximum velocity of the object (in cm/s)?

The position of a mass oscillating on a spring is given by x = ( 3.6 cm) cos[2πt/(0.67s)].A. What is the period of this motion?T=? sB. What is the first time the mass is at the position x = 0?t=? s

(Figure 1) is the position-versus-time graph of a particle in simple harmonic motion.(a) What is the phase constant?(b) What is the velocity at t=0s?(c) What is vmax?

A 200g oscillator in a vacuum chamber has a frequency of 2hz. When air is admitted, the oscillation decreases to 60% of its initial amplitude in 50s How many oscillations will have been completed when the amplitude is 30% of its initial value?

Figure 1 shows a harmonic oscillator at four different moments, labeled A, B, C, and D. Assume that the force constant k, the mass of the block, m, and the amplitude of vibrations, A, are given. We will also assume that there are no resistive forces so the total energy of the oscillator remains constant.Find the kinetic energy of the block at the moment labeled B.

Introduction to OscillationsThe conditions that lead to simple harmonic motion are as follows:There must be a position of stable equilibrium.There must be a restoring force acting on the oscillating object. The direction of this force must always point toward the equilibrium, and its magnitude must be directly proportional to the magnitude of the object's displacement from its equilibrium position. Mathematically, the restoring force F⃗ is given by F⃗ = −kx⃗, where x⃗ is the displacement from equilibrium and k is a constant that depends on the properties of the oscillating system.The resistive forces in the system must be reasonably small.Consider a block of mass m attached to a spring with force constant k, as shown in the figure (Figure 1). The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located at x=0. If the block is pulled to the right a distance A and then released, A will be the amplitude of the resulting oscillations.Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block.Now assume for the remaining Parts G -J, that the x coordinate of point R is 0.12 m and the t coordinate of point K is 0.0050 s.What distance d does the object cover between the moments labeled K and N on the graph? Express your answer in meters.

Introduction to OscillationsThe conditions that lead to simple harmonic motion are as follows:There must be a position of stable equilibrium.There must be a restoring force acting on the oscillating object. The direction of this force must always point toward the equilibrium, and its magnitude must be directly proportional to the magnitude of the object's displacement from its equilibrium position. Mathematically, the restoring force F⃗ is given by F⃗ = −kx⃗, where x⃗ is the displacement from equilibrium and k is a constant that depends on the properties of the oscillating system.The resistive forces in the system must be reasonably small.Consider a block of mass m attached to a spring with force constant k, as shown in the figure (Figure 1). The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located at x=0. If the block is pulled to the right a distance A and then released, A will be the amplitude of the resulting oscillations.Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block.Now assume for the remaining Parts G -J, that the x coordinate of point R is 0.12 m and the t coordinate of point K is 0.0050 s.What distance d does the object cover during one period of oscillation? Express your answer in meters.

Introduction to OscillationsThe conditions that lead to simple harmonic motion are as follows:There must be a position of stable equilibrium.There must be a restoring force acting on the oscillating object. The direction of this force must always point toward the equilibrium, and its magnitude must be directly proportional to the magnitude of the object's displacement from its equilibrium position. Mathematically, the restoring force F⃗ is given by F⃗ = −kx⃗, where x⃗ is the displacement from equilibrium and k is a constant that depends on the properties of the oscillating system.The resistive forces in the system must be reasonably small.Consider a block of mass m attached to a spring with force constant k, as shown in the figure (Figure 1). The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located at x=0. If the block is pulled to the right a distance A and then released, A will be the amplitude of the resulting oscillations.Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block.The time it takes the block to complete one cycle is called the period. Usually, the period is denoted T and is measured in seconds.The frequency, denoted f, is the number of cycles that are completed per unit of time: f = 1/T. In SI units, f is measured in inverse seconds, or hertz (Hz).An oscillating object takes 0.10 s to complete one cycle; that is, its period is 0.10 s. What is its frequency f? Express your answer in hertz.

Introduction to OscillationsThe conditions that lead to simple harmonic motion are as follows:There must be a position of stable equilibrium.There must be a restoring force acting on the oscillating object. The direction of this force must always point toward the equilibrium, and its magnitude must be directly proportional to the magnitude of the object's displacement from its equilibrium position. Mathematically, the restoring force F⃗ is given by F⃗ = −kx⃗, where x⃗ is the displacement from equilibrium and k is a constant that depends on the properties of the oscillating system.The resistive forces in the system must be reasonably small.Consider a block of mass m attached to a spring with force constant k, as shown in the figure (Figure 1). The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located at x=0. If the block is pulled to the right a distance A and then released, A will be the amplitude of the resulting oscillations.Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block.The time it takes the block to complete one cycle is called the period. Usually, the period is denoted T and is measured in seconds.The frequency, denoted f, is the number of cycles that are completed per unit of time: f = 1/T. In SI units, f is measured in inverse seconds, or hertz (Hz).If the period is doubled, the frequency isA. unchanged.B. doubled.C. halved.

What is the frequency of the oscillation shown in the figure?

For the graph below, determine the frequency f and the oscillation amplitude A

a) What is the amplitude of the oscillation shown in (Figure 1)?b) What is the frequency of the oscillation shown in the figure?

The figure shows a light extended spring exerting a force Fs to the left on a block.(i) Does the block exert a force on the spring? Select all that apply.a) Its magnitude is equal to Fs.b) Yes, to the right.c) Its magnitude is larger than Fs.d) Yes, to the left.e) No, it does not.f) Its magnitude is smaller than Fs.(ii) Does the spring exert a force on the wall? Select all that apply.a) Its magnitude is smaller than Fs.b) Yes, to the right.c) Its magnitude is equal to Fs.d) Its magnitude is larger than Fs.e) No, it does not.f) Yes, to the left.

What is the phase constant of the oscillation shown in the figure? Suppose that −180° ≤ ϕ ≤180°.

Discuss how a "restoring force" and an "equilibrium position" are related?

Introduction to Oscillations0 Advanced issue found▲ Figure 10 Advanced issue found▲ Figure 2The conditions that lead to simple harmonic motion are as follows:There must be a position of stable equilibrium.There must be a restoring force acting on the oscillating object. The direction of this force must always point toward the equilibrium, and its magnitude must be directly proportional to the magnitude of the object's displacement from its equilibrium position. Mathematically, the restoring force F⃗ is given by F⃗ = −kx⃗, where x⃗ is the displacement from equilibrium and k is a constant that depends on the properties of the oscillating system.The resistive forces in the system must be reasonably small.Consider a block of mass m attached to a spring with force constant k, as shown in the figure (Figure 1). The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located at x=0. If the block is pulled to the right a distance A and then released, A will be the amplitude of the resulting oscillations.Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block.After the block is released from x = A, it willA. remain at rest.B. move to the left until it reaches equilibrium and stop there.C. move to the left until it reaches x = −A and stop there.D. move to the left until it reaches x=−A and then begin to move to the right.

A block oscillating on a spring has period T = 2.0sa) What is the period if the block's mass is doubled? Explain.b) What is the period if the value of the spring constant is quadrupled?c) What is the period if the oscillation amplitude is doubled while m and k are unchanged?

Spring constant problem. What is the formula for k in terms of the period T and the mass?A. K= 4π2m/T2B. K= 4π2/mT2C. K= 2πm/T

Figure 1 shows a harmonic oscillator at four different moments, labeled A, B, C, and D. Assume that the force constant k, the mass of the block, m, and the amplitude of vibrations, A, are given. We will also assume that there are no resistive forces so the total energy of the oscillator remains constant.What is the maximum potential energy of the system?

Figure 1 shows a harmonic oscillator at four different moments, labeled A, B, C, and D. Assume that the force constant k, the mass of the block, m, and the amplitude of vibrations, A, are given. We will also assume that there are no resistive forces so the total energy of the oscillator remains constant.What is the total energy of the system?

Introduction to OscillationsThe conditions that lead to simple harmonic motion are as follows:There must be a position of stable equilibrium.There must be a restoring force acting on the oscillating object. The direction of this force must always point toward the equilibrium, and its magnitude must be directly proportional to the magnitude of the object's displacement from its equilibrium position. Mathematically, the restoring force F⃗ is given by F⃗ = −kx⃗, where x⃗ is the displacement from equilibrium and k is a constant that depends on the properties of the oscillating system.The resistive forces in the system must be reasonably small.Consider a block of mass m attached to a spring with force constant k, as shown in the figure (Figure 1). The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located at x=0. If the block is pulled to the right a distance A and then released, A will be the amplitude of the resulting oscillations.Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block.Which points on the x axis are located a distance A from the equilibrium position?A. R onlyB. Q onlyC. both R and Q

A machine part is undergoing SHM with a frequency of 5.00Hz and amplitude 1.80cm . How long does it take the part to go from x=0 to x= -1.80 cm?

Figure 1 shows a harmonic oscillator at four different moments, labeled A, B, C, and D. Assume that the force constant k, the mass of the block, m, and the amplitude of vibrations, A, are given. We will also assume that there are no resistive forces so the total energy of the oscillator remains constant. Which moment corresponds to the maximum kinetic energy of the system? You can indicate the answer by the letter A, B, C, or D.

Figure 1 shows a harmonic oscillator at four different moments, labeled A, B, C, and D. Assume that the force constant k, the mass of the block, m, and the amplitude of vibrations, A, are given. We will also assume that there are no resistive forces so the total energy of the oscillator remains constant.Which moment corresponds to the minimum kinetic energy of the system? You can indicate the answer by the letter A, B, C, or D.

You can double the maximum speed of an object on a spring undergoing simple harmonic motion by:a. Reducing the mass to one-fourth its original valueb. All of thesec. Doubling the amplituded. None of thesee. Increasing the spring constant to four times its original value

For a simple harmonic oscillator, answer yes or no to the following questions.(a) Can the quantities position and velocity have the same sign? Yes No (b) Can velocity and acceleration have the same sign? Yes No (c) Can position and acceleration have the same sign? Yes No

(Figure 1) shows a position-versus-time graph for a particle in SHM.A) What is the amplitude A?B) What is the angular frequency ω?C) What is the phase constant ϕ0?

Figure 1 shows a harmonic oscillator at four different moments, labeled A, B, C, and D. Assume that the force constant k, the mass of the block, m, and the amplitude of vibrations, A, are given. We will also assume that there are no resistive forces so the total energy of the oscillator remains constant.Which moment corresponds to the maximum potential energy of the system? You can indicate the answer by the letter A, B, C, or D.

Figure 1 shows a harmonic oscillator at four different moments, labeled A, B, C, and D. Assume that the force constant k, the mass of the block, m, and the amplitude of vibrations, A, are given. We will also assume that there are no resistive forces so the total energy of the oscillator remains constant.Which moment corresponds to the minimum potential energy of the system? You can indicate the answer by the letter A, B, C, or D.

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