Oscillation
1. Let’s review some basic concepts of mechanical oscillation.
Solution:
Oscillation is a motion in which a body, after being displaced, passes through its equilibrium position several times. In this position, the oscillating body has the lowest potential energy. Oscillation is characterized by quantities:
2.Determine the period and frequency
- a.) of the sewing machine needle, which makes 20 stitches per second
- b.) of the heartbeat, which performs 75 beats per minute
Solution:
- Needle: f = 20 Hz, T = 0.05 s.
- Heart: f = 1.25 Hz, T = 0.8 s.
3. The harmonic oscillation of an oscillator is described by the equation y = 8.sin(4πt + 0.25π) cm. Determine:
- a.) the amplitude of displacement
- b.) the period
- c.) the frequency
Calculate the instantaneous displacement at time t = 0s
Solution:
The instantaneous displacement at time t = 0s is y = 5.656 cm.
4.The oscillation of an oscillator is described by the equation y = 4.sin(0.5πt) cm. After what time will the instantaneous displacement of the oscillator reach half the amplitude?
Solution:
The oscillator reaches the instantaneous displacement after t = 1/3 seconds.
5. How does the period of harmonic oscillation change if, instead of a copper cylinder (ρ1 = 8930 kg.m-3), we attach an aluminum cylinder (ρ2 = 2700 kg.m-3) of the same volume to the spring?
Solution:
The period of oscillation decreases by about 0.55 times.
6. In the cabin of an elevator hangs a pendulum with a period T1 = 1s. When the cabin moves with constant acceleration, the pendulum oscillates with a period T2 = 1.2s. Determine the acceleration of the cabin.
Solution:
The cabin has acceleration a = –3 m.s-2.
7.The period of natural oscillation of a railway wagon is 1.25s. Due to impacts at rail joints, the wagon receives force impulses that set it into oscillation. At what train speed will the wagon resonate the most, if the length of the rails is 25 m?
Solution:
Analysis:
T = 1.25s, s = 25m, v = ?
The wagon resonates the most at speed v = 72 km.h-1
8.A material point oscillates harmonically with amplitude ym = 5 cm, period T = 2s, and φ = 0o. Determine the velocity of the point at the moment when the instantaneous displacement is 2.5 cm.
Solution:
Analysis:
The velocity of the point is v = 0.14 m.s-1
9.A material point performs harmonic motion described by the equation y = 5.sin(6πt) cm. At what time is its kinetic energy three times greater than its potential energy?
Solution:
The required energy ratio occurs at time t = 1/36 s.
10. The total energy of a harmonic oscillator is 3·10-5 J and the maximum force acting on it is 1.5·10-3 N. Write the equation of instantaneous displacement if the oscillator has a period T = 2s and an initial phase φ = 60o.
Solution:
11.What is the fastest way to get from Europe to Australia?
Solution:
Dig a tunnel from Europe to Australia through the center of the Earth! If you fall into the shaft in Europe, you will oscillate between Europe and Australia. The travel time there and back is the period of this oscillatory motion.
