Energy Balance Reading Page Energy in Motion
Simple Harmonic Motion & Energy Conservation
Physics Reference Guide
Name: ___________
Date: ___________
The Energy Identity
In a system exhibiting Simple Harmonic Motion (SHM) , the total mechanical energy (\(E_{total}\)) is the sum of the system's kinetic energy (\(K\)) and potential energy (\(U\)).
\(E_{total} = K + U\)
Conservation of Energy
The total mechanical energy remains constant throughout the cycle. While \(K\) and \(U\) exchange values as the object moves, their sum never fluctuates in an ideal system.
Key Takeaway
"Energy is never lost in an ideal SHM system; it simply dances between motion and position."
Vertical Mass-Spring System
Equilibrium Position
Max Compression
m
\(+A\)
\(U_{max}\) | \(K = 0\)
Equilibrium
m
\(U_{min}\) | \(K_{max}\)
Max Extension
m
\(-A\)
\(U_{max}\) | \(K = 0\)
Horizontal Mass-Spring System
Equilibrium Line
Compressed
m
Restoring Force (\(F\))
Displacement (\(x = -A\))
Equilibrium
m
Net Force = 0
Maximum Kinetic Energy
Stretched
m
Restoring Force (\(F\))
Displacement (\(x = +A\))
Quick Check: Energy Balance
Total Energy is constant in SHM.
Force points toward equilibrium.
Potential is max at amplitudes (\(\pm A\)).
Kinetic is max at equilibrium.
Energy Analysis Practice Sheet Energy Analysis
SHM Practice & Application
Name: ___________
Date: ___________
Part 1: System Identification
A) Energy Transformations
Indicate whether the energy is at its MAX , MIN (0) , or NEITHER .
Position
Kinetic Energy (K)
Potential Energy (U)
Total Energy (E)
Equilibrium (\(x = 0\))
Positive Amplitude (\(+A\))
Negative Amplitude (\(-A\))
Midway Point (\(x = A/2\))
B) Kinematics & Dynamics Comparison
Compare values using MAX , MIN (0) , or DIRECTION (+/-) indicators.
Position
Displacement
Force
Acceleration
Velocity
Equilibrium
Max (\(+A\))
Min (\(-A\))
Turning Point
Part 2: Conservation Calculations
1. A mass-spring system (\(k = 200 \text{ N/m}\)) is stretched to an amplitude of \(0.1 \text{ m}\). Calculate the total mechanical energy of the system.
2. What is the kinetic energy of the mass as it passes through equilibrium? Explain using conservation of energy.
3. If the amplitude is tripled to \(0.3 \text{ m}\), what is the new total mechanical energy? By what factor has it increased?
Part 3: System Shifts
A vertical mass-spring system is oscillating. If you double the mass (\(2m\)) but keep the same amplitude , how does the total mechanical energy change? Explain.
Energy Exchange Slides Energy in Motion
Conservation in Simple Harmonic Systems
Unit 7: SHM AP Physics 1
The Energy Identity
Total mechanical energy is the sum of a system's motion and its position .
\[E_{total} = K + U\]
Kinetic Energy (\(K\))
Energy of motion. Depends on mass and speed.
Potential Energy (\(U\))
Stored energy. Depends on displacement from equilibrium.
Equilibrium Position
Vertical System Analysis
Compressed
m
\(+A\)
U is MAX
K = 0
Equilibrium
m
U is MIN
K is MAX
Stretched
m
\(-A\)
U is MAX
K = 0
Equilibrium Reference
Horizontal System Analysis
COMPRESSED
\(x = -A\)
m
Restoring Force
U is MAX
EQUILIBRIUM
\(x = 0\)
m
Net Force = 0
K is MAX
STRETCHED
\(x = +A\)
m
Restoring Force
U is MAX
Visualizing Balance
- Amplitude Equilibrium + Amplitude
Potential Energy (\(U\))
Kinetic Energy (\(K\))
Total Energy (\(E\))
The Rule of Squares
How does the Total Energy react to a change in Amplitude ?
\[E_{total} \propto A^2\]
Double Amplitude (\(2A\))
4x Energy
Triple Amplitude (\(3A\))
9x Energy
Energy Analysis Teacher Guide Teacher Guide
Energy in Motion: Simple Harmonic Motion
Subject: AP Physics 1
Learning Objectives
Explain kinetic vs. potential energy exchange in SHM.
Connect kinematic variables (x, v, a) with restoring force.
Identify energy states at key positions (equilibrium/amplitude).
Common Misconceptions
Mass Trap: For a fixed amplitude, energy depends only on \(k\) and \(A\).Linear thinking: Doubling amplitude quadruples energy (\(E \propto A^2\)).
Answer Key: Energy Analysis
Part 1A: Energy Transformations
Position K U Total E Equilibrium MAX MIN (0) CONSTANT Amplitude (+A) MIN (0) MAX CONSTANT Amplitude (-A) MIN (0) MAX CONSTANT Midway (A/2) NEITHER NEITHER CONSTANT
Part 1B: Kinematics & Dynamics
Position Disp. Force Accel. Veloc. Equilibrium 0 0 0 MAX Max (\(+A\)) \(+A\) MAX (-) MAX (-) 0 Min (\(-A\)) \(-A\) MAX (+) MAX (+) 0 Turning Point \(\pm A\) MAX MAX 0
Part 2: Calculations
1. \(1.0\text{ J}\) | 2. \(1.0\text{ J}\) (K is max where U is min) | 3. \(9.0\text{ J}\) (Quadrupled)
Part 3: Shifts
Total energy remains unchanged . Energy capacity in SHM is defined solely by the spring's stiffness (\(k\)) and the extent of compression/stretch (\(A\)). Mass only influences the timing (period) of the exchange.
