SIMPLE HARMONIC MOTION AP/Honors Physics 1

Mr. Velazquez

Periodic Motion

• Periodic Motion refers to any motion that repeats over and over • Beating heart, pendulum, sound waves, etc.

• One of the key features of any periodic motion is the time it takes to complete one cycle of motion, known as the period. It is measured in units of seconds (s).

• Closely related to the period is its reciprocal, known as the frequency, or the number of cycles completed in one second. This is measured in units of Hertz (s−1). • These units were named after German physicist Heinrich Hertz, because of

his crucial work on radio waves.

Period: 𝑇 = time required for one cycle of motion

Frequency:

𝑓 =1

𝑇 (number of cycles in one second)

Simple Harmonic Motion

• One special type of periodic motion is called simple harmonic motion

• Suppose we suspend a spring from the ceiling and attach a mass to the end of the spring. Then we stretch or compress the spring a bit and let it go.

• This apparatus is called a spring pendulum, and the resulting motion comes from the periodic oscillation of the spring.

Simple Harmonic Motion • When the spring is stretched or compressed, Hooke’s law tells us that

there is a reaction force in the spring equal to 𝐹 = −𝑘𝑥 • When it is released, potential energy from the spring is converted to

kinetic energy when it reaches equilibrium length. This kinetic energy then gets transferred into elastic potential energy again at the other extreme, and the process will repeat again when it reaches the point where it started.

• The absolute value of this extreme displacement is known as the amplitude of motion.

Simple Harmonic Motion (Position)

We can use the following equation to represent the displacement or position 𝑥 of the mass at any time 𝑡:

𝜔 =2𝜋

𝑇= 2𝜋𝑓 𝒙 = 𝑨 𝒄𝒐𝒔 𝝎𝒕

Note: Make sure that your calculator is set to RADIANS when using these formulas.

Angular Frequency Position in SHM

Simple Harmonic Motion (Velocity)

The velocity of the mass will vary at different positions, but it is greatest at the equilibrium point, and zero at the extremes. This can be represented algebraically using the following equations:

𝜔 =2𝜋

𝑇= 2𝜋𝑓 𝒗 = −𝑨𝝎 𝐬𝐢𝐧(𝝎𝒕)

Note: Make sure that your calculator is set to RADIANS when using these formulas.

Angular Frequency Velocity in SHM

Simple Harmonic Motion (Acceleration)

Applying the same principles as we used for the velocity, we can also define the acceleration of the object using reference points; it is zero at equilibrium, and alternates between positive and negative at the extrema:

𝜔 =2𝜋

𝑇= 2𝜋𝑓 𝐚 = −𝑨𝝎𝟐 𝐜𝐨𝐬(𝝎𝒕)

Note: Make sure that your calculator is set to RADIANS when using these formulas.

Angular Frequency Acceleration in SHM

Simple Harmonic Motion

A vertical spring pendulum is lifted 10 cm above its equilibrium point and released so that it begins to oscillate with simple harmonic motion. Assuming it takes 1.45 s to return to the same height it started from:

a) Write the equations representing the position, velocity and acceleration for the simple harmonic motion of the pendulum

b) Find the displacement of the pendulum after 7.00 s.

Exit Ticket Part 1: Equations for SHM A spring pendulum is observed oscillating with simple harmonic motion. The pendulum completes exactly 3.50 oscillations per second, and the mass at the end reaches a maximum height of 5.35 cm above the equilibrium line. Assuming it’s at this maximum height at time 𝑡 = 0:

a) Write the equations for the position, velocity and acceleration of the mass

b) Use these equations to find the exact position, velocity and acceleration of the mass at time 𝑡 = 9.36 s.

𝒙 = 𝑨 𝒄𝒐𝒔 𝝎𝒕

𝒗 = −𝑨𝝎 𝐬𝐢𝐧(𝝎𝒕)

𝐚 = −𝑨𝝎𝟐 𝐜𝐨𝐬(𝝎𝒕)

𝜔 =2𝜋

𝑇= 2𝜋𝑓

Period for a Spring Pendulum Recall the equation for the net force on a spring:

𝐹 = −𝑘𝑥 Now we substitute 𝐹 = 𝑚𝑎:

𝑚𝑎 = −𝑘𝑥 Then, substituting the equations we just obtained for position and acceleration:

𝑚 −𝐴𝜔2 cos 𝜔𝑡 = −𝑘(𝐴 cos(𝜔𝑡)) 𝑚𝜔2 = 𝑘

𝜔 =𝑘

𝑚

Substituting 𝜔 = 2𝜋/𝑇 gives us the final equation:

𝑻 = 𝟐𝝅𝒎

𝒌 Period of a Spring

Pendulum

To obtain the period for a simple pendulum, we use a combination of force equations and rotational motion equations. (For a detailed description of this, see pgs. 401-403 in your book)

In short, the period for a simple pendulum is almost identical to that of a spring pendulum, except the restoring force 𝑘 is replaced with 𝑚𝑔/𝐿:

Period of a Simple Pendulum

Period of a Simple Pendulum 𝑻 = 𝟐𝝅

𝑳

𝒈

• A physical pendulum is one where mass is NOT concentrated at a single point at the end of the pendulum, but instead is distributed over some finite volume.

• Some examples would include a clock pendulum, a human femur, or a thick heavy rope.

Period of a Physical Pendulum

Period of a Physical Pendulum 𝑻 = 𝟐𝝅

𝑳

𝒈

𝑰

𝒎𝑳𝟐

• Calculating the period for a physical pendulum requires that we know the moment of inertia (𝐼), total mass (𝑚), and length (𝐿; to the center of mass) of the pendulum.

𝐿

Exit Ticket Part 2: SHM and Pendulums

To build a clock, you are asked to find a pendulum with a period of exactly 1.00 seconds, and you would like to offer two options: a spring pendulum (using a spring with a k-constant of 950 N/m) or simple pendulum. Find:

a) The mass required for the spring pendulum

b) The length required for the simple pendulum

Period of a Simple Pendulum

𝑻 = 𝟐𝝅𝑳

𝒈

Period of a Spring Pendulum

𝑻 = 𝟐𝝅𝒎

𝒌

AP: (due 2/12) SHM Packet

Honors: (due 2/5)

Pg. 413-415 #4-56 (mult of 4)

Homework