Chapter 11-2 Measuring Simple Harmonic Motion St. Augustine Preparatory School March 31, 2016

Definitions Amplitude: The maximum displacement from equilibrium (meters or radians) Period: The time that it takes a complete cycle to occur (unit: seconds) Frequency: The number of cycles or vibrations per unit time (unit: Hertz (s-1) )

The period of a pendulum depends on which factors?

The period of a pendulum depends on which factors? Pendulum Length and Free Fall Acceleration (gravity)

Period of a Simple Pendulum in Simple Harmonic Motion Formula: Period = 2*pi*square root of (length/acceleration of gravity) T: Period (unit: s) 2π: 2*3.14 (unitless) L = length (unit: m) ag = acceleration of gravity (unit: m/s2). Use 9.81 m/s2 on Earth

Nevis Swing – New Zealand! https://www.youtube.com/watch?v=Ux0tKX2TXOc (start at 2:20)

Example 1 – Nevis Swing Queenstown, New Zealand is the home of the World’s Largest Swing. Here, you are connected to a 120. m long rope (a soccer field is 110. m long) which makes an arc with a distance of 300 m. The site claims you travel at a speed of 120 km/h on this swing. Does this claim seem reasonable? Why or why not? (This will require us to think about a few different things).

Example 2 You jump off of a platform with a rope around feet and end up in a parabolic path, like a pendulum. If your period for one cycle is 12 s, how high above the ground is the platform you are swinging from?

Solution

Frequency The formula for frequency is: Frequency is equal to 1 divided by the period f = frequency (unit: Hz) T = period (unit: s)

Period for Mass Spring Systems The period of a mass spring system depends on: Mass Spring Constant Why does mass matter now when it didn’t in the pendulum system?

Increasing the mass of an object increases the inertia (resistance to movement) of that object . In a pendulum, increasing the mass also increases the Fg on the object, so the increase in mass is compensated for. In a spring system it does not add any extra force, so the mass does matter.

Period for Mass Spring Systems in SHM 𝑇=2𝜋 𝑚 𝑘 Period = 2*pi*sqrt(mass divided by spring constant) Period – seconds (s) Mass – kilograms (kg) Spring Constant – Newtons per meter (N/m)

Practice Problem The body of a 1275 kg car is supported on a frame by four springs. Two people riding in the car have a combine mass of 153 kg. When driven over a pothole in the road, the frame vibrates with a period of 0.840 s. For the first few seconds, the vibration approximates simple harmonic motion. Find the spring constant of a single spring.

Solution

Graded Problems A 125 N object vibrates with a period of 3.56 s when hanging from a spring. What is the spring constant of the spring? You are designing a pendulum clock to have a period of 1.0 s. How long should the pendulum be? A trapeze artist swings in simple harmonic motion with a period of 3.8 s. Calculate the length of the cables supporting the trapeze. Calculate the period and frequency of a 3.500 m long pendulum at Jakarta, Indonesia, where the acceleration from gravity is 9.782 m/s2.