Math Problem Statement
An object oscillates in simple harmonic motion, reaching a maximum velocity of 1.2 m/s whenever it passes through the central position, which occurs every 3.6 s.
-find the amplitude and maximum acceleration of the harmonic motion.
Solution
To find the amplitude and maximum acceleration of the object in simple harmonic motion, we'll use the following formulas related to simple harmonic motion (SHM):
-
Maximum Velocity Formula: where:
- is the maximum velocity,
- is the angular frequency,
- is the amplitude.
-
Relation Between Angular Frequency and Period: where:
- is the period of the motion.
-
Maximum Acceleration Formula:
Step-by-Step Solution:
-
Determine Angular Frequency ():
Given that the period s, the angular frequency is:
-
Find the Amplitude ():
Using the maximum velocity formula: Plugging in and : Solving for :
-
Calculate Maximum Acceleration ():
Using the maximum acceleration formula: Substituting the values of and :
Final Answers:
- Amplitude (): Approximately .
- Maximum Acceleration (): Approximately .
Would you like more details or have any questions?
Related Questions:
- How does the period of the motion affect the amplitude and maximum velocity?
- What would happen to the maximum acceleration if the amplitude were doubled?
- How can the energy of the system be calculated using the amplitude and mass?
- How would the maximum velocity change if the period was reduced to half?
- What is the phase of the object when it is at its maximum displacement?
Tip:
In simple harmonic motion, the maximum velocity occurs at the equilibrium position, while the maximum acceleration occurs at the endpoints of the motion.
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Math Problem Analysis
Mathematical Concepts
Simple Harmonic Motion
Angular Frequency
Amplitude
Acceleration
Formulas
v_max = ωA
ω = 2π / T
a_max = ω^2A
Theorems
Simple Harmonic Motion Theorem
Suitable Grade Level
Grades 10-12
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