15.7
Problem Solving: Energy in Simple Harmonic Motion
Simple harmonic motion (SHM) is a type of periodic motion in time and position, in which an object oscillates back and forth around an equilibrium pos…
Consider a car moving over a bumpy road. Here, the suspension of the car acts like a spring connecting the mass of the car to the wheels.
During its motion, the spring compresses and expands, executing simple harmonic motion.
Suppose the car bounces with a certain amplitude; what is its maximum velocity, and how does its total energy change if it is halfway between its initial and equilibrium positions?
The total energy in simple harmonic motion remains conserved. Rearrangement of terms gives the velocity at any position during the motion.
The maximum velocity occurs when the car's spring passes through the equilibrium position. On substituting the known quantities, the maximum velocity is obtained. The total energy at the equilibrium position is then obtained by substituting the known and calculated quantities.
When the car's spring moves from maximum to half amplitude, the velocity is considered negative. The total energy is calculated by substituting the velocity with other known quantities.
In both cases, the total energy of simple harmonic motion is the same.
View the full transcript and gain access to JoVE Core videos
Q1: Why does total energy remain constant in simple harmonic motion?
Total energy in simple harmonic motion remains conserved because the force on the spring is conservative. Energy continuously exchanges between potential energy stored when the spring compresses or extends and kinetic energy during motion. At every point during oscillation, the sum of potential and kinetic energy stays constant, provided no dissipative forces act on the system.
Q2: At what point during oscillation does maximum velocity occur?
Maximum velocity occurs when the car's spring passes through the equilibrium position. At this point, potential energy stored in the spring is zero, so all total energy converts to kinetic energy. This is where the oscillating object moves fastest during its back-and-forth motion around the equilibrium position.
Q3: How does total energy change when an oscillating object is halfway between maximum and equilibrium positions?
Total energy remains the same at all positions, including halfway between maximum and equilibrium positions. When the object moves from maximum amplitude toward equilibrium, potential energy decreases while kinetic energy increases proportionally. The total mechanical energy is proportional to the square of the amplitude and stays constant throughout the motion.
Q4: What relationship does the energy conservation equation reveal in simple harmonic motion?
The energy conservation equation presents a useful relationship between velocity, position, and total mechanical energy without requiring time. This equation is valuable when problems require relations between position, velocity, and acceleration independent of time. It allows students to solve oscillation problems by analyzing energy exchanges rather than tracking motion through time intervals.
Q5: How can you determine the signs of displacement and velocity in simple harmonic motion problems?
Signs of displacement and velocity must be inferred from the physical situation described. If the body moves from equilibrium toward the greatest positive displacement, both displacement and velocity are positive. Conversely, when moving from maximum displacement back toward equilibrium, displacement remains positive but velocity becomes negative, indicating direction reversal.
Q6: Why is studying energy in simple harmonic motion important for engineering applications?
Understanding energy in simple harmonic motion is vital for analyzing oscillating systems in physics and engineering, such as car suspension springs. Energy analysis reveals how systems behave under different conditions without solving complex time-dependent equations. This knowledge helps engineers design shock absorbers and other mechanical systems that must control oscillations effectively.
Q7: How does the spring in a car's shock absorber demonstrate energy conservation principles?
The spring attached to a car's wheel executes simple harmonic motion while the car moves on a bumpy road. As the spring compresses and extends, potential energy stores in the compressed or extended spring, then converts to kinetic energy as the wheel oscillates. This continuous energy exchange demonstrates that total mechanical energy remains conserved throughout the oscillation cycle.