Solving Simple Harmonic Motion Problems in Physics

Problem 1: Horizontal Spring and Mass System

• Given:
  • Mass (m) = 0.75 kg
  • Spring constant (k) = 300 N/m
• Objective: Calculate the period (T), frequency (f), and angular frequency (ω).

Calculations:

• Period (T):

  • Formula: ( T = 2\pi \sqrt{\frac{m}{k}} )
  • Calculation: ( T = 2\pi \sqrt{\frac{0.75}{300}} = 0.3142 ) seconds

• Frequency (f):

  • Formula: ( f = \frac{1}{T} )
  • Calculation: ( f = \frac{1}{0.3142} = 3.183 ) Hz

• Angular Frequency (ω):

  • Formula: ( \omega = 2\pi f )
  • Calculation: ( \omega = 2\pi \times 3.183 = 20 ) radians/second

Problem 2: Force Applied to Stretch a Spring

• Given:
  • Force (F) = 500 N
  • Mass (m) = 0.5 kg
  • Displacement (x) = 0.35 m
• Objective: Calculate the spring constant (k) and frequency (f).

Calculations:

• Spring Constant (k):

  • Formula: ( F = kx )
  • Calculation: ( k = \frac{500}{0.35} = 1428.6 ) N/m

• Frequency (f):

  • Formula: ( f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} )
  • Calculation: ( f = \frac{1}{2\pi} \sqrt{\frac{1428.6}{0.5}} = 8.51 ) Hz

Problem 3: Frequency Change with Spring Constant

• Given:
  • Initial Spring constant (k1) = 100 N/m
  • Initial Frequency (f1) = 25 Hz
  • New Spring constant (k2) = 400 N/m
• Objective: Calculate the new frequency (f2).

Calculations:

• Proportional Change:
  • Relationship: ( f \propto \sqrt{k} )
  • ( \sqrt{\frac{k2}{k1}} = \sqrt{\frac{400}{100}} = 2 )
  • Calculation: ( f2 = 2 \times 25 = 50 ) Hz

Problem 4: Vibrating Mass with Given Equation

• Given:
  • Mass (m) = 0.75 kg
  • Displacement Equation: ( x = 0.65 \cos(7.35t) )
• Objective: Determine amplitude, frequency, period, and spring constant.

Calculations:

• Amplitude (A):

  • Value from Equation: 0.65

• Angular Frequency (ω):

  • From Equation: 7.35 rad/s

• Frequency (f):

  • Formula: ( \omega = 2\pi f )
  • Calculation: ( f = \frac{7.35}{2\pi} = 1.17 ) Hz

• Period (T):

  • Formula: ( T = \frac{1}{f} )
  • Calculation: ( T = \frac{1}{1.17} = 0.855 ) seconds

• Spring Constant (k):

  • Formula: ( k = m \omega^2 )
  • Calculation: ( k = 0.75 \times (7.35)^2 = 40.5 ) N/m

Key Concepts

• Period (T) is the time taken for one complete cycle in harmonic motion.
• Frequency (f) is the number of cycles per second.
• Angular Frequency (ω) relates to how quickly the oscillations occur in radians per second.
• Spring Constant (k) measures the stiffness of the spring.
• Amplitude (A) is the maximum displacement from the equilibrium position.

This lecture provided a comprehensive understanding of how to approach and solve various simple harmonic motion problems using key physics formulas.