Mass-Spring System Dynamics: Energy, Period, and Frequency

Total Mechanical Energy Conservation: In Simple Harmonic Motion (SHM), the total mechanical energy is conserved and alternates between potential and kinetic forms.
Elastic Potential Energy ( $EPE$ ): Reaches its maximum at the amplitudes ( $x = A$ and $x = -A$ ). The formula is $EPE = \frac{1}{2} k x^2$ .
Kinetic Energy ( $KE$ ): Reaches its maximum at the equilibrium position ( $x = 0$ ) and its minimum at the amplitudes. The formula is $KE = \frac{1}{2} m v^2$ .
Velocity: The maximum speed ( $v_{max}$ ) is achieved when all energy is kinetic ( $KE_{max} = EPE_{max}$ ).

Period ( $T$ ): The time for one full oscillation is determined by the formula $T = 2\pi \sqrt{\frac{m}{k}}$ .
Frequency ( $f$ ): The number of oscillations per second is determined by the formula $f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$ .

System 1: A mass of $1.4\,kg$ and spring constant ( $k$ ) of $75.0\,N/m$ stretched by $5.0\,cm$ ( $0.050\,m$ ).
System 2: A mass of $0.50\,kg$ and $k = 120.0\,N/m$ used to find oscillation period.
System 3: A mass of $2.00\,kg$ and $k = 125.0\,N/m$ used to find frequency.
System 4: A mass of $2.0\,kg$ , $k = 200\,N/m$ , and amplitude ( $A$ ) of $3.0\,m$ . * Total Energy: $E = \frac{1}{2} (200\,N/m)(3.0\,m)^2 = 900\,J$ . * Maximum Velocity: $v_{max} = 30\,m/s$ .
System 5: A mass of $4.0\,kg$ and $k = 64\,N/m$ . * Period: $1.6\,s$ . * Frequency: $0.64\,Hz$ .

Q23: At which location(s) is the kinetic energy a maximum? Answer: $x = 0$ .
Q24: At which location(s) is the spring potential energy a maximum? Answer: $x = A$ and $x = -A$ .
Q25: At which location(s) is the total energy a maximum? Answer: All locations (energy is conserved).
Q26: At which location(s) is the kinetic energy a minimum? Answer: $x = A$ and $x = -A$ .
Q27-30: These slides involve calculating energy ( $900\,J$ ), velocity ( $30\,m/s$ ), period ( $1.6\,s$ ), and frequency ( $0.64\,Hz$ ) for specific mass-spring parameters provided as practice examples.