by Dr ejaz ahmad

1. PHY-401
Waves and Oscillations
BS Physics
4th Semester
Dr. Ejaz Ahmed
Department of Physics
Abdul Wali Khan University Mardan

2. Lecture 4: Topics to be Covered
1. Displacement, Velocity and Acceleration
2. Potential Energy, Kinetic Energy and Total Energy

3. Solution of 2nd Order Differential Equation
If the wave advances by a time 𝑻 = Τ
𝟐𝝅
𝝎 = 𝟐𝝅
𝒎
𝒌
the cosine function repeats
itself. This is called the time period of the oscillation i.e., the time in which the wave
repeats itself. Its unit is in sec.
𝑥(𝑡) = 𝐴 cos( 𝜔𝑡 + 𝜙)
𝝎 = 𝒂𝒏𝒈𝒖𝒍𝒂𝒓 𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚 ൗ
𝒓𝒂𝒅𝒊𝒂𝒏𝒔
𝒔𝒆𝒄
𝑨 = 𝒂𝒎𝒑𝒍𝒊𝒕𝒖𝒅𝒆 𝒎 , for cosine function maximum displacement is ±1.
𝝓 = 𝒑𝒉𝒂𝒔𝒆 𝒂𝒏𝒈𝒍𝒆 𝒓𝒂𝒅𝒊𝒂𝒏𝒔
The frequency of the scalation is 𝒇 = Τ
𝟏
𝑻, so 𝒇 = Τ
𝝎
𝟐𝝅and it is in Hz.
𝝎𝒕 + 𝝓 = 𝒂𝒏𝒈𝒍𝒆

4. Position of Particle at any Time in Simple Harmonic Motion
𝑥(𝑡) = 𝐴 cos( 𝜔𝑡 + 𝜙)

5. Velocity in Simple Harmonic Motion
𝑥(𝑡) = 𝐴 cos( 𝜔𝑡 + 𝜙)
𝑣(𝑡) =
𝑑𝑥(𝑡)
𝑑𝑡
𝑣(𝑡) = −𝜔𝐴 sin( 𝜔𝑡 + 𝜙)
=
𝑑[𝐴 cos( 𝜔𝑡 + 𝜙)]
𝑑𝑡

6. Acceleration in Simple Harmonic Motion
𝑥(𝑡) = 𝐴 cos( 𝜔𝑡 + 𝜙)
𝑎 𝑡 =
𝑑𝑣 𝑡
𝑑𝑡
𝑎(𝑡) = −𝜔2
𝑥(𝑡)
= −𝜔2
𝐴 cos( 𝜔𝑡 + 𝜙)
=
𝑑2
𝑥(𝑡)
𝑑𝑡2

7. Comparison of the Three Quantities

8. Total Energy in Simple Harmonic Motion
In an Isolated system, where there are no dissipative forces, the total mechanical energy remain constant
i.e., Conserved.
Total Energy= Potential Energy + Kinetic Energy
𝐸 = 𝑈 + 𝐾
Where E is Total Energy, U is Potential Energy and K is the Kinetic Energy
Potential Energy:
As we know that 𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 = 𝑤𝑜𝑟𝑘 = 𝑑𝑈 = −𝐹. 𝑑𝑥
𝑈(𝑥) = − න
0
𝑥𝑚
𝐹. 𝑑𝑥 = − න
0
𝑥𝑚
−𝑘𝑥 . 𝑑𝑥 = 𝑘 න
0
𝑥𝑚
𝑥. 𝑑𝑥 =
1
2
𝑘 𝑥𝑚
2
𝑈 𝑥 =
1
2
𝑘 𝑥𝑚
2
=
1
2
𝑘 𝑥𝑚
2
𝑐𝑜𝑠2
𝜔𝑡 + 𝜙

9. Total Energy in Simple Harmonic Motion
Kinetic Energy:
As we know that
where
𝐾 =
1
2
𝑚 𝑣𝑥
2
=
1
2
𝑚 𝜔2 𝑥𝑚
2 𝑠𝑖𝑛2 𝜔𝑡 + 𝜙
𝐾 =
1
2
𝑘 𝑥𝑚
2
𝑠𝑖𝑛2
𝜔𝑡 + 𝜙
𝜔2 =
𝑘
𝑚
and 𝑚𝜔2
= 𝑘

10. Total Energy in Simple Harmonic Motion
Total Energy= Potential Energy + Kinetic Energy
𝑬 = 𝑼 + 𝑲
Putting values from previous work
𝐸 =
1
2
𝑘 𝑥𝑚
2 𝑐𝑜𝑠2 𝜔𝑡 + 𝜙 +
1
2
𝑘 𝑥𝑚
2 𝑠𝑖𝑛2 𝜔𝑡 + 𝜙
𝐸 =
1
2
𝑘 𝑥𝑚
2
𝑐𝑜𝑠2
𝜔𝑡 + 𝜙 + 𝑠𝑖𝑛2
𝜔𝑡 + 𝜙
𝐸 =
1
2
𝑘𝑥𝑚
2
Figure: The potential energy 𝑼, kinetic energy
𝑲, and total mechanical energy 𝑬 of a particle
undergoing simple harmonic motion (with 𝝓 =
𝟎) are shown as functions of (𝒂) the time and
(𝒃) the displacement. Note that in (𝒂) the kinetic
and potential energies each reach their maxima
twice during each period of the motion.
(a)
(b)

11. Figure: Simple harmonic motion for a block–spring system and its relationship to the motion of a simple pendulum.
The parameters in the table refer to the block–spring system, assuming that 𝑥 = 𝐴 at t = 0 thus, 𝑥 = 𝐴𝑐𝑜𝑠𝜔𝑡

12. Video Lectures to watch:
1. https://www.youtube.com/watch?v=tNpuTx7UQbw (8.01x- Lect 10 - Hooke's Law, Springs,
Pendulums, Simple Harmonic Motion)
2. https://www.youtube.com/watch?v=GOdrEXTkWyI&list=PLIKpuUo6d5pIy313eSfqgHxx1P2cd5O7
n&index=181 (Energy in SHM)
3. https://www.youtube.com/watch?v=SmuCOSwuoOs&list=PLIKpuUo6d5pIy313eSfqgHxx1P2cd5O
7n&index=182 (Dynamics of SHM)