## Energy in Simple Harmonic Motion

13.2 Energy in Simple Harmonic Motion

### Learning Objectives

By the end of this section, you will be able to:

- Describe the energy conservation of the system of a mass and a spring
- Explain the concepts of stable and unstable equilibrium points

**Practice!**

Practice 13.2.1 |
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A horizontal spring has one end firmly attached to a wall and the other end attached to a mass. The mass can slide freely on a smooth, horizontal surface with no friction. The mass is pulled away from the equilibrium position by a distance A in the positive x-direction and then released. At which position is the kinetic energy of the mass-spring system a maximum? |

(a) The kinetic energy is zero throughout the motion. |

(b) The kinetic energy is constant throughout the motion. |

(c) At x = A |

(d) At x = 0 |

(e) At x = A/2 |

The kinetic energy of the mass-spring system is a maximum **at x = 0**. The total energy of the mass-spring system is the sum of the kinetic energy of the mass and the potential energy in the spring. Energy is conserved throughout the motion, with the energy transferring back and forth between the kinetic energy of the mass and the potential energy in the spring. The kinetic energy of the mass will be a maximum value when the potential energy in the spring is zero, which occurs at

*x*= 0, the equilibrium position. This is because the spring is neither stretched nor compressed when the system is at equilibrium, which makes the potential energy in the spring equal to zero.

When the mass is at the position *x* = *A*, the spring is stretched by a maximum amount and the mass is at rest for an instant. This means the potential energy in the spring is a maximum value and the kinetic energy of the mass is zero.

Practice 13.2.2 |
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A horizontal spring has one end firmly attached to a wall and the other end attached to a mass. The mass can slide freely on a smooth, horizontal surface with no friction. The mass is pulled away from the equilibrium position by a distance A in the positive x-direction and then released. At which position is the potential energy of the mass-spring system a maximum? |

(a) The potential energy is zero throughout the motion. |

(b) The potential energy is constant throughout the motion. |

(c) At x = A |

(d) At x = 0 |

(e) At x = A/2 |

The potential energy of the mass-spring system is a maximum **at x = A**. The total energy of the mass-spring system is the sum of the kinetic energy of the mass and the potential energy in the spring. Energy is conserved throughout the motion, with the energy transferring back and forth between the kinetic energy of the mass and the potential energy in the spring. When the mass is at the position

*x*=

*A*, the spring is stretched by a maximum amount and the mass is at rest for an instant. This means the potential energy in the spring is a maximum value and the kinetic energy of the mass is zero.

The kinetic energy of the mass will be a maximum value when the potential energy in the spring is zero, which occurs at *x* = 0, the equilibrium position. This is because the spring is neither stretched nor compressed when the system is at equilibrium, which makes the potential energy in the spring equal to zero.

**Discuss!**

This is an x-t graph for an object connected to a spring and moving in simple harmonic motion.

(a) At which of the following times is the **potential** energy of the spring the greatest?

(b) At which of the following times is the **kinetic** energy of the spring the greatest?

A 0.550 kg cart connected to a light spring for which the force constant is 22.5 N/m oscillates on a horizontal, frictionless air track. When the mass is at the center and at rest, we give it a kick and increase its velocity to 0.305 m/s.

(a) What is the amplitude (in cm) of the oscillation?

(b) When the mass is at -2.5 cm, what is its speed (in m/s)?

When displaced from equilibrium by a small amount, the two hydrogen atoms in an H_{2} molecule are acted on by a restoring force F_{x }= −k_{1}x with k_{1} = 510 N/m.

Calculate the oscillation frequency f of the H_{2} molecule. Use m_{eff }= m/2 as the “effective mass” of the system, where m is the mass of a hydrogen atom. Take the mass of a hydrogen atom as 1.008 u, where 1 u = 1.661 × 10^{−27} kg. Express your answer in hertz.

A rifle bullet with mass 7.00 g and initial horizontal velocity 280 m/s strikes and embeds itself in a block with mass 0.993 kg that rests on a frictionless surface and is attached to one end of an ideal spring. The other end of the spring is attached to the wall. The impact compresses the spring a maximum distance of 18.0 cm. After the impact, the block moves in SHM.

Calculate the period of this motion.