## Pendulums

13.4 Pendulums

### Learning Objectives

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

- State the forces that act on a simple pendulum
- Determine the angular frequency, frequency, and period of a simple pendulum in terms of the length of the pendulum and the acceleration due to gravity
- Define the period for a physical pendulum

### Simple Pendulum

A **simple pendulum** is defined to have a point mass, also known as the pendulum bob, which is suspended from a string of length *L* with negligible mass. The only forces acting on the bob are the force of gravity (i.e., the weight of the bob) and tension from the string. The mass of the string is assumed to be negligible as compared to the mass of the bob.

**Practice!**

Practice 13.4.1 |
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A simple pendulum consists of a point mass, m, attached to the end of a massless string of length L. It is pulled out of its straight-down equilibrium position by a small angle [insert: \theta] and released so that it oscillates about the equilibrium position in simple harmonic motion. A graph showing the pendulum’s angular position as a function of time is given in the figure. What is the frequency of the pendulum’s motion? |

(a) 1.6 Hz |

(b) 2.0 Hz |

(c) 0.33 Hz |

(d) 3.0 Hz |

(e) 0.63 Hz |

The frequency of the pendulum’s motion is **0.63 Hz**. The frequency is defined as the number of cycles per unit time and can be calculated with the inverse of the period: *f* = 1/*T*. The period of the pendulum’s motion, which is the time it takes the mass to move from some initial position back to that initial position, is 1.6 s. The frequency is then 1/1.6 s, which equals 0.63 Hz.

Practice 13.4.2 |
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A simple pendulum consists of a point mass, m, attached to the end of a massless string of length L. It is pulled out of its straight-down equilibrium position by a small angle [insert: \theta] and released so that it oscillates about the equilibrium position in simple harmonic motion. A graph showing the pendulum’s angular position as a function of time is given in the figure. What is the period of the pendulum’s motion? |

(a) 3.0 s |

(b) 6 s |

(c) 2.0 s |

(d) 1.6 s |

(e) 1.0 s |

The period of the pendulum’s motion is **1.6 s**. The period is defined as the time to complete one cycle, or one round-trip. To measure the period, you need to measure the time it takes the mass to move from some initial position back to that initial position. The initial angular position is 6 degrees and the pendulum returns to this position after 1.6 s have elapsed.

Practice 13.4.3 |
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You are sitting on a swing. A friend gives you a small push and you start swinging back & forth with period T _{1}. Suppose you were standing on the swing rather than sitting. When given a small push you start swinging back & forth with period T_{2}.Which of the following is true: |

(a) T_{1} = T_{2} |

(b) T_{1} > T_{2} |

(c) T_{1} < T_{2} |

Practice 13.4.4 |
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Consider a pendulum, which has a period of 1 second on the Earth. What will its period be on the moon? |

(a) 1 s |

(b) 1.8 s |

(c) 2.4 s |

(d) 3.2 s |

(e) 6 s |

### Physical Pendulum

A **physical pendulum** is any object whose oscillations are similar to those of the simple pendulum, but cannot be modeled as a point mass on a string, and the mass distribution must be included into the equation of motion.

**Practice!**

Practice 13.4.5 |
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What length do we make the simple pendulum so that it has the same period as the rod pendulum? |

(a) L_{S} = (3/2)L_{R} |

(b) L_{S} = (2/3)L_{R} |

(c) L_{S} = L_{R} |

Practice 13.4.6 |
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What is the period of a meter stick if you suspend it from one end and allow it to swing in simple harmonic motion? |

(a) 1 s |

(b) 1.6 s |

(c) 2 s |

(d) 2.6 s |

(e) 3 s |

Pause & Predict 13.4.1 |
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What is the acceleration due to gravity on Mars? |

(a) 4.39 m/s^{2} |

(b) 1.85 m/s^{2} |

(c) 13.8 m/s^{2} |

(d) 1.43 m/s^{2} |

(e) 3.71 m/s^{2} |

Pause & Predict 13.4.2 |
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How far away from the pivot is the pendulum’s center of mass? |

(a) 0.73 m |

(b) 0.26 m |

(c) 0.52 m |

(d) 0.083 m |

(e) 0.17 m |

Practice 13.4.7 |
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A baseball bat is 32 inches (81.3 cm) long and has a mass of 0.96 kg. Its center of mass is 22 inches (55.9 cm) from the handle end. You hold the bat at the very tip of the handle end (the knob) and let it swing in simple harmonic motion. What is the bat’s moment of inertia if its period of oscillation is 1.35 seconds? |

(a) 0.353 kg•m^{2} |

(b) 0.243 kg•m^{2} |

(c) 1.13 kg•m^{2} |

(d) 9.55 kg•m^{2} |

**Discuss!**

A meter stick swings as a pendulum, with an axis of rotation 0.250 meters from the end. What is its period? (Hint: use the parallel-axis theorem)

A pendulum consists of a sphere of radius r = 1.0 cm tied to a light string so that the center of mass of the sphere is a distance L = 49.4 cm from the pivot point.

(a) What is the period of this physical pendulum?

(b) What error is made in calculating the period by treating the sphere as a point mass?