71. Where Is the Energy? A block of mass m is attached to a...

71. Where Is the Energy? A block of mass $m$ is attached to a spring of spring constant $k$ that is attached to a wall as shown in Fig. $16-49 .$ (a) If the block starts at time $t=0$ with the spring being $\quad$ FiGURE $16-49=$ at its rest length but the block having a $\quad$ Problem 71 . velocity $v_{1}$, find a solution for the mass's position at all subsequent times. Make any assumptions you like in order to have a plausible but solvable model, but state your assumptions explicitly. (b) Are the energies at the times $$ t=0, \quad t=\frac{\pi}{2} \sqrt{\frac{m}{k}}, \quad t=2 \pi \sqrt{\frac{m}{k}} $$ kinetic, potential, or a mixture of the two? (This is not a shortanswer question. Show how you know.)

71. Where Is the Energy? A block of mass $m$ is attached to a spring of spring constant $k$ that is attached to a wall as shown in Fig. $16-49 .$ (a) If the block starts at time $t=0$ with the spring being $\quad$ FiGURE $16-49=$ at its rest length but the block having a $\quad$ Problem 71 . velocity $v_{1}$, find a solution for the mass's position at all subsequent times. Make any assumptions you like in order to have a plausible but solvable model, but state your assumptions explicitly. (b) Are the energies at the times
$$
t=0, \quad t=\frac{\pi}{2} \sqrt{\frac{m}{k}}, \quad t=2 \pi \sqrt{\frac{m}{k}}
$$
kinetic, potential, or a mixture of the two? (This is not a shortanswer question. Show how you know.)

Key Concepts

Potential Energy

Potential energy in a spring-mass system is stored in the spring when it is either compressed or stretched. This energy, described by a quadratic function of the displacement, reaches its maximum when the mass is at the extremes of its motion, where the velocity is zero and all energy is stored as potential energy.

Initial Conditions

Initial conditions specify the starting state of a dynamical system, including quantities like initial displacement and velocity. They are crucial for solving differential equations uniquely, enabling one to determine the specific trajectory of the system over time.

Differential Equations

The motion of an oscillating system, such as a spring-mass system, is modeled by a second-order linear differential equation. Solving this equation with appropriate initial conditions yields the time-dependent behavior of the system, including amplitude, phase, and period of motion.

Kinetic Energy

Kinetic energy is the energy associated with the motion of the mass in the system. It is determined by the mass and the square of its velocity. In oscillatory systems, the kinetic energy reaches its maximum when the mass passes through the equilibrium position where the velocity is highest.

Hooke's Law

Hooke's Law is a principle that states that the force exerted by a spring is proportional to its displacement from the equilibrium position, and acts in the opposite direction. It provides the basis for understanding the restoring force in spring-mass systems, and is essential for deriving the equations governing simple harmonic motion.

Conservation of Energy

This principle asserts that in an isolated system, energy is neither created nor destroyed but can be transformed from one form to another. In the context of oscillatory systems, it describes how the total mechanical energy (sum of kinetic and potential energies) remains constant, even though the energy may periodically shift between kinetic energy when the mass is moving and potential energy when the spring is compressed or extended.

Simple Harmonic Motion

This concept describes periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction. It is typically modeled by a second?order differential equation and characterizes systems that oscillate about an equilibrium position with a constant angular frequency dependent on the system's parameters.

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