⏱️ Simple Harmonic Motion Calculator
Calculate SHM parameters
How to Use This Calculator
Enter Amplitude
Input the amplitude (A) in meters (m). This is the maximum displacement from equilibrium - the farthest distance the object moves from its center position. For example, if a spring oscillates 0.1 m on each side of equilibrium, enter 0.1.
Enter Frequency
Input the frequency (f) in hertz (Hz). This is how many complete oscillations occur per second. For example, if a pendulum completes 2 full swings per second, enter 2. Higher frequency means faster oscillation.
Enter Time
Input the time (t) in seconds at which you want to calculate the motion parameters. This is the specific moment in the oscillation cycle you're interested in. For example, enter 0.5 to see the position after half a second.
Calculate and Review
Click the "Calculate" button to compute displacement, velocity, and acceleration at the specified time. The calculator also shows the period and angular frequency. Displacement is measured from equilibrium, velocity is the rate of change of position, and acceleration is the rate of change of velocity.
Formula
x(t) = A sin(ωt)
v(t) = Aω cos(ωt)
a(t) = -Aω² sin(ωt)
ω = 2πf, T = 1/f
Where:
• x(t) = Displacement at time t (m)
• v(t) = Velocity at time t (m/s)
• a(t) = Acceleration at time t (m/s²)
• A = Amplitude (m)
• ω = Angular frequency (rad/s)
• f = Frequency (Hz)
• t = Time (s)
Example 1: Calculating at Specific Time
A spring-mass system oscillates with amplitude 0.2 m and frequency 2 Hz. Calculate displacement, velocity, and acceleration at t = 0.25 s.
Given:
• Amplitude (A) = 0.2 m
• Frequency (f) = 2 Hz
• Time (t) = 0.25 s
Solution:
ω = 2πf = 2π × 2 = 4π rad/s
x(t) = 0.2 × sin(4π × 0.25) = 0.2 × sin(π) = 0 m
v(t) = 0.2 × 4π × cos(4π × 0.25) = 0.8π × cos(π) = -2.51 m/s
a(t) = -0.2 × (4π)² × sin(4π × 0.25) = -0.2 × 16π² × sin(π) = 0 m/s²
At t = 0.25 s: x = 0 m, v = -2.51 m/s, a = 0 m/s²
Example 2: Maximum Values
What are the maximum velocity and acceleration for an SHM system with amplitude 0.1 m and frequency 5 Hz?
Given:
• Amplitude (A) = 0.1 m
• Frequency (f) = 5 Hz
Solution:
ω = 2πf = 2π × 5 = 10π rad/s
v_max = Aω = 0.1 × 10π = π ≈ 3.14 m/s
a_max = Aω² = 0.1 × (10π)² = 0.1 × 100π² ≈ 98.7 m/s²
Maximum velocity: 3.14 m/s, Maximum acceleration: 98.7 m/s²
Frequently Asked Questions
What is simple harmonic motion (SHM)?
Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to the displacement. The motion follows a sinusoidal (sine or cosine) pattern. Examples include: a mass on a spring, a pendulum with small angles, and many oscillating systems in physics.
What's the relationship between displacement, velocity, and acceleration in SHM?
In SHM, displacement, velocity, and acceleration are all sinusoidal functions but are out of phase: When displacement is maximum, velocity is zero and acceleration is maximum (but negative). When displacement is zero, velocity is maximum and acceleration is zero. Velocity is the derivative of displacement, and acceleration is the derivative of velocity.
Why does the acceleration formula have a negative sign?
The negative sign indicates that acceleration always opposes displacement - this is the restoring force characteristic of SHM. When the object is displaced to the right, acceleration points left (negative), pulling it back toward equilibrium. This is what creates the oscillatory motion.
What happens at different times in the cycle?
At t = 0: Typically starts at equilibrium (x = 0) with maximum velocity. At t = T/4: Maximum displacement, zero velocity, maximum acceleration. At t = T/2: Back to equilibrium, maximum velocity (opposite direction). At t = 3T/4: Maximum displacement (opposite side), zero velocity. The pattern repeats every period T.
How does frequency affect the motion?
Higher frequency means faster oscillation (shorter period) and higher maximum velocity and acceleration for the same amplitude. Since velocity v_max = Aω and ω = 2πf, doubling the frequency doubles the maximum velocity. Since acceleration a_max = Aω², doubling frequency quadruples the maximum acceleration.
Where is simple harmonic motion used in real-world applications?
SHM is fundamental in: pendulum clocks and timekeeping devices, spring-mass systems (shock absorbers, suspension systems), musical instruments (vibrating strings, air columns), AC circuits (oscillating current and voltage), atomic physics (electron motion in atoms), and many mechanical and electrical oscillators. Understanding SHM is essential for analyzing any periodic motion.
About Simple Harmonic Motion Calculator
The simple harmonic motion calculator computes displacement, velocity, and acceleration for objects in simple harmonic motion using sinusoidal functions. SHM is one of the most important types of periodic motion in physics.
This calculator is essential for students studying oscillatory motion, engineers designing oscillating systems, and anyone analyzing periodic phenomena. It helps you understand how position, velocity, and acceleration vary with time in harmonic motion, providing insights into the dynamics of oscillating systems from pendulums to springs to AC circuits.