Embedded Files
From the College Board (AP)
From the College Board (AP)
2.2 The student can apply mathematical routines to quantities that describe natural phenomena.
4.2 The student can design a plan for collecting data to answer a particular scientific question.
5.1 The student can analyze data to identify patterns or relationships.
6.2 The student can construct explanations of phenomena based on evidence produced through scientific practices.
6.4 The student can make claims and predictions about natural phenomena based on scientific theories and models.
7.2 The student can connect concepts in and across domain(s) to generalize or extrapolate in and/or across enduring understandings and/or big ideas.
LEARNING OBJECTIVE
3.B.3.1 Predict which properties determine the motion of a simple harmonic oscillator and what the dependence of the motion is on those properties. [SP 6.4, 7.2]
3.B.3.2 Design a plan and collect data in order to ascertain the characteristics of the motion of a system undergoing oscillatory motion caused by a restoring force. [SP 4.2]
3.B.3.3 Analyze data to identify qualitative and quantitative relationships between given values and variables (i.e., force, displacement, acceleration, velocity, period of motion, frequency, spring constant, string length, mass) associated with objects in oscillatory motion and use those data to determine the value of an unknown. [SP 2.2, 5.1]
Construct a qualitative and/ or quantitative explanation of oscillatory behavior given evidence of a restoring force. [SP 2.2, 6.2]
Restoring forces can result in oscillatory motion. When a linear restoring force is exerted on an object displaced from an equilibrium position, the object will undergo a special type of motion called simple harmonic motion. Examples include gravitational force exerted by Earth on a simple pendulum and mass-spring oscillator.
a. For a spring that exerts a linear restoring force, the period of a mass-spring oscillator increases with mass and decreases with spring stiffness.
b. For a simple pendulum, the period increases with the length of the pendulum and decreases with the magnitude of the gravitational field.
c. Minima, maxima, and zeros of position, velocity, and acceleration are features of harmonic motion. Students should be able to calculate force and acceleration for any given displacement for an object oscillating on a spring.
Skills:
1.C - Create qualitative sketches of graphs that represent features of a model or the behavior of a physical system.
2.A - Derive a symbolic expression from known quantities by selecting and following a logical mathematical pathway.
2.B - Calculate or estimate an unknown quantity with units from known quantities, by selecting and following a logical computational pathway.
3.B - Apply an appropriate law definition, theoretical relationship, or model to make a claim.
Key Terms:
Key Terms:
Amplitude (A): The maximum displacement of an oscillating object from its equilibrium position.
Angular frequency (ω): A measure of how rapidly an object oscillates, expressed in radians per second.
Conservation of energy: The principle that the total energy of an isolated system remains constant.
Displacement (x): The distance and direction of an object from its equilibrium position.
Elastic potential energy (PE): The energy stored in a stretched or compressed spring.
Equilibrium position: The position where an object experiences zero net force.
Frequency (f): The number of oscillations per unit time, measured in Hertz (Hz).
Hooke's Law: The law stating that the force exerted by a spring is proportional to its displacement from equilibrium (F = -kx).
Kinetic energy (KE): The energy of motion.
Mass-spring system: A system consisting of a mass attached to a spring that can oscillate.
Natural frequency: The frequency at which a system oscillates when disturbed.
Overdamped: A type of damping where the system returns to equilibrium slowly without oscillating.
Period (T): The time taken for one complete oscillation.
Pendulum: A mass suspended from a fixed point that can swing back and forth.
Periodic motion: Motion that repeats itself in equal intervals of time.
Restoring force: A force that brings an object back towards its equilibrium position.
Resonance: The phenomenon that occurs when a system is driven at its natural frequency, resulting in large amplitude oscillations.
Simple harmonic motion (SHM): A type of periodic motion where the restoring force is proportional to the displacement and always directed towards the equilibrium position.
SHM Overview:
SHM Overview:
Oscillations play a great part in our lives, from the tides to the motion of the swinging pendulum that once governed our perception of time. General principles govern this area of physics, from water waves in the deep ocean or the oscillations of a car suspension system. This introduction to the topic reminds us that not all oscillations are isochronous. However, the simple harmonic oscillator is of great importance to physicists because all periodic oscillations can be described through the mathematics of simple harmonic motion.
Mass spring systems are typically orientated either vertically or horizontally. The vertically orientated springs are easy to experiment with in a lab setting, however the horizontally orientated spring systems are difficult to replicate due to the frictional forces involved.
Some questions to consider from the graph above:
How does changing the initial spring stretch (Δx) change the initial energy of the system?
How does changing the mass of the block (m) affect the initial energy of the system?
Desribe the relationship amongst the position (x), velocity (v) and acceleration (a) graphs.
Outline why the potential energy (U_s) and kinetic energy (K) appear to have a period of 1/2 that of the position graph.
Good practice for those enrolled in AP Calc.
Complete the following document to summarize the motion of a mass on a spring.
MOS description.pdfOne Cycle:
One Cycle:
Necessary Equations
Necessary Equations
A 24 cm spring has a spring / force constant (k) of 400 N/m. How much force is required to strech the spring to a length of 28 cm?
Vertical Mass on a Spring
Vertical Mass on a Spring
Lecture Notes HERE
A spring of force constant 1600 N/m is mounted on a horizontal table as shown in the figure. A mass m= 4.0 kg attached to the free end of the spring is pulled horizontally towards the right through a distance of 4.0 cm and then set free.
Calculate the frequency
maximum acceleration
maximum speed of the mass.
Period v. Frequency
Period v. Frequency
Motion of a Pendulum
Motion of a Pendulum
Some questions to consider from the graph above:
How does changing the initial pendulum length (Δx) change the initial energy of the system?
How does changing the mass of the block (m) affect the initial energy of the system?
Desribe the relationship amongst the position (x), velocity (v) and acceleration (a) graphs.
Outline why the potential energy (U_s) and kinetic energy (K) appear to have a period of 1/2 that of the position graph.
Pendulum Description.pdfPage updated
Google Sites
Report abuse