66. Oscillating Graphs A mass is hanging from a spring off...

66. Oscillating Graphs A mass is hanging from a spring off the edge of a table. The position of the mass is measured by a sonic ranger sitting on the floor $25 \mathrm{~cm}$ below the mass's equilibrium position. At some time, the mass is started oscillating. At a later time, the sonic ranger begins to take data. Figure $16-47$ shows a series of graphs associated with the motion of the mass and a series of physical quantities. The graph labeled $(\mathrm{A})$ is a graph of the mass's position as measured by the ranger. For each physical quantity, identify which graph could represent that quantity for this situation. If none are possible, answer $\mathrm{N}$. (a) Velocity of the mass (b) Net force on the mass (c) Force exerted by the spring on the mass (d) Kinetic energy of the spring-mass system (e) Potential energy of the spring-mass-Earth system (f) Gravitational potential energy of the spring-mass-Earth system

66. Oscillating Graphs A mass is hanging from a spring off the edge of a table. The position of the mass is measured by a sonic ranger sitting on the floor $25 \mathrm{~cm}$ below the mass's equilibrium position. At some time, the mass is started oscillating. At a later time, the sonic ranger begins to take data.
Figure $16-47$ shows a series of graphs associated with the motion of the mass and a series of physical quantities. The graph labeled $(\mathrm{A})$ is a graph of the mass's position as measured by the ranger. For each physical quantity, identify which graph could represent that quantity for this situation. If none are possible, answer $\mathrm{N}$.
(a) Velocity of the mass
(b) Net force on the mass
(c) Force exerted by the spring on the mass
(d) Kinetic energy of the spring-mass system
(e) Potential energy of the spring-mass-Earth system
(f) Gravitational potential energy of the spring-mass-Earth system

Key Concepts

Kinetic Energy in Oscillatory Systems

Kinetic energy in a mass-spring system depends on the square of the velocity. It reaches its maximum when the mass is moving fastest (typically at the equilibrium position) and drops to zero when the mass momentarily stops at the turning points. Its non-negative nature and dependence on the square of a sinusoidal function lead to a graph that may show a doubled frequency compared to the displacement graph.

Hooke’s Law

This law states that the force exerted by an ideal spring is proportional to the displacement from its equilibrium position and acts in the opposite direction. It is a foundational principle in the analysis of spring-mass systems, explaining how the spring force contributes to the net force on the mass during oscillation.

Potential Energy in Oscillatory Systems

Potential energy in such systems can include both the elastic potential energy stored in the spring and the gravitational potential energy due to the mass's height. The energy changes as the system oscillates, with energy converting between kinetic and potential forms. The elastic potential energy specifically depends on the square of the displacement, while gravitational potential energy varies linearly with height.

Simple Harmonic Motion

This concept refers to oscillatory motion where the restoring force acting on a system is directly proportional to its displacement from equilibrium. As a result, the system follows a sinusoidal motion characterized by a constant amplitude and period, which is fundamental in understanding how displacement, velocity, and acceleration vary over time.

Phase Relationships

In simple harmonic motion, physical quantities such as displacement, velocity, and acceleration (or net force) are out of phase with each other. For example, the velocity reaches its maximum when the displacement is zero, while the acceleration, which is proportional to the net force, is in phase with the displacement but opposite in direction. These phase shifts are key to interpreting oscillatory graphs.

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