7.2 Mass-Spring Systems

• What is Simple Harmonic Motion (SHM)?

  Periodic motion with restoring force proportional to displacement
• In SHM, the acceleration is directly proportional to the displacement.


  True
• What are the three key components of a mass-spring system?

  Mass, spring, equilibrium position
• In SHM, the restoring force is directly proportional to the displacement
• What is the equation for the restoring force in SHM?

  $F =$$- kx$
• The restoring force in SHM is given by $F =$$- kx$, where $k$ is the spring constant and $x$ is the displacement
• The period of oscillation in a mass-spring system is given by $T =$$2\pi\sqrt{\frac{m}{k}}$, where m</latex> is the mass and $k$ is the spring constant
• Hooke's Law states that the restoring force is proportional to the displacement from equilibrium.

  True
• The spring in a mass-spring system provides the restoring force
• What is the equilibrium position in a mass-spring system?

  Point where the spring is unstretched
• In SHM, the restoring force increases as the displacement from equilibrium increases.

  True
• What is the equation for the period of oscillation in SHM?

  $T =$$2\pi\sqrt{\frac{m}{k}}$
• The frequency of oscillation in SHM is given by $f =$$\frac{1}{T}$, where $T$ is the period
• What type of energy is maximum when the mass in a mass-spring system is at the equilibrium position?

  Kinetic energy
• The total mechanical energy in a mass-spring system undergoing SHM remains constant if no dissipative forces are present.

  True
• The potential energy stored in a spring is given by $U =$$\frac{1}{2}kx^{2}$, where $k$ is the spring constant and $x$ is the displacement
• The total mechanical energy of a system remains constant if no energy is lost to friction
• What happens to kinetic energy as a mass moves away from the equilibrium position in SHM?

  Decreases
• What is the role of the spring in a mass-spring system?

  Provides restoring force
• The total mechanical energy of a mass-spring system is the sum of kinetic and potential energies


  True
• Match the position in a mass-spring system with its energy characteristics:

  Equilibrium ↔️ Maximum KE, Minimum PE

  Maximum Displacement ↔️ Minimum KE, Maximum PE
• In a mass-spring system undergoing SHM, kinetic energy is maximum at the equilibrium
• What is the formula for potential energy in a mass-spring system?

  $U =$$\frac{1}{2} kx^{2}$
• The spring constant (k) measures the stiffness of the spring in a mass-spring system.


  True
• What is the period of oscillation in a mass-spring system defined as?

  Time for one complete cycle
• Steps to derive the period of oscillation for a mass-spring system:

  1️⃣ Start with the acceleration equation for SHM

  2️⃣ Rewrite as a differential equation

  3️⃣ Find the general solution

  4️⃣ Relate period to angular frequency

  5️⃣ Substitute angular frequency into the period equation
• In the period equation, \omega = \sqrt{\frac{k}{m}}</latex> represents the angular frequency
• The period of oscillation is inversely proportional to the angular frequency.


  True
• What is the final equation for the period of oscillation in a mass-spring system?

  $T =$$2\pi\sqrt{\frac{m}{k}}$
• Match the variable with its description in the period equation:

  T ↔️ Period of oscillation

  m ↔️ Mass

  k ↔️ Spring constant
• Increasing the mass of a mass-spring system increases its period of oscillation.


  True
• Match the factor with its effect on the period of oscillation:

  Increase in mass ↔️ Increases period

  Increase in spring constant ↔️ Decreases period
• The period of oscillation depends on the mass and the spring constant
• Match the factor with its impact on the period of oscillation:

  Mass ↔️ Increases

  Spring Constant ↔️ Decreases
• What is the equation for the restoring force in SHM?

  $F =$$- kx$
• Match the property with its equation in SHM:

  Restoring Force ↔️ $F =$$- kx$

  Period ↔️ $T =$$2\pi\sqrt{\frac{m}{k}}$

  Frequency ↔️ $f =$$\frac{1}{T}$
• What characterizes the restoring force in a mass-spring system?

  Spring constant
• The equilibrium position in a mass-spring system is the point where the spring is unstretched
• Hooke's Law states that the restoring force is proportional to the displacement from equilibrium.


  True
• Order the energy transformations in a mass-spring system during one oscillation cycle:

  1️⃣ Mass at equilibrium position with maximum kinetic energy

  2️⃣ Mass moves away from equilibrium, kinetic energy decreases, potential energy increases

  3️⃣ Mass at maximum displacement with maximum potential energy

  4️⃣ Mass returns to equilibrium, potential energy decreases, kinetic energy increases