How do astronauts weigh themselves in space? The answer to that question and more are in Episode 22.
How do astronauts weigh themselves in space? The answer to that question and more are in Episode 22. Because masses on springs oscillate, we start by relating the period of a spring to the factors affecting it. (1:16) Brief time is spent explaining the difference between period and frequency, and we define many of the key terms associated with the wave nature of oscillations. (2:43) The equation to find position of an oscillating object is provided and utilized to determine the velocity of the object for a given time and various other sinusoidal relationships are discussed as well. (4:27)
The Question of the Day asks: (9:23)
How can the period of mass on a spring be doubled?
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Hi and welcome to the APsolute RecAP: Physics 1 Edition. Today’s episode will focus on spring oscillators.
Let’s Zoom out:
Unit 6 – Simple Harmonic Motion
Topic 6.1-6.2
Big Idea – Force Interactions and Conservation
Have you ever wondered, how do astronauts get weighed? If you don’t know, space agencies are very knowledgeable about the health of any space crew. They need to be. One basic health metric is your mass or weight, but how can a space agency measure this quantity when the crew members are falling around the planet in an orbit? Oscillating springs, that’s how!
Let’s Zoom in:
Go ahead and find your trusty Physics 1 equation sheet. On the sheet, locate the equations for simple harmonic motion. They will both be on the left hand side and will have a 2(Pi) in them. The one equation is the period of a pendulum, the other is the period of a spring oscillator. You should recall that for a pendulum, the acceleration due to gravity is related, but for the spring oscillator it is not a factor. Mass however - is, and so is the spring constant. In the United States, the space agency is titled the National Aeronautics and Space Administration, or NASA for short. Maybe you’ve heard of it? They’re kind of a big deal. One way that NASA astronauts measure their mass is to ride a pogo-stick like machine that oscillates them back and forth. This records the period of their oscillations and determines astronaut mass using a spring with a known spring constant.
Let’s try it out. The equation for the period of a spring oscillator is 2(pi)multiplied by the square root of the quantity m/k. If NASA measures the period of oscillations for an astronaut to be 2.9 s and they know the spring’s stiffness constant is 350 N/m, then they can accurately determine the astronauts mass to be about 75 kg or 165 lbs.
Just a reminder, the period is the time for one single oscillation. However, you may want to know what’s called the frequency of an oscillating object. It is just the inverse concept. Instead of number of seconds for a single oscillation, frequency is the number of oscillations that occur in a single second. Also, because they are inverse concepts, it is super easy to convert between period and frequency. In the case of our astronaut, they oscillated with a period of 2.9 s for each oscillation, so 1/(2.9) will show that the astronauts frequency was 0.34 oscillation per second. Or, 0.34 Hertz. Easy!
As a mass oscillates, it would be nice to know the position of the oscillating mass at any given moment. As you can imagine, that may be difficult to measure if the object is constantly in motion. Luckily, we have an equation that can tell us the position of a mass that is oscillating with a given frequency. First, we should pick a location for the origin, and most people pick that to be the mass’ equilibrium position, or where it has a net force of zero acting on it. If there is zero net force, then the mass also cannot be accelerating at that position. Since Newton’s 2nd Law is in fact, well, a law - we can’t violate it. I like to imagine that this is a horizontal spring and that when no force is applied, the coils have some room to be compressed. This is opposed to one of those tightly wound tension springs that can only be stretched. The equilibrium will therefore lie in the middle of each oscillation, so there will be times when the position is positive, and other times when it is negative. When the mass on the spring is at its maximum displacement, in either the positive or negative direction, we call that location the amplitude. On a position vs time graph, the amplitude will be the maximum vertical deviation from the origin, or equilibrium position.
Now that you know the basics, the formula on the equation sheet will make some sense. This is the one that reads x = Acos(2𝝅ft) where capital A is the amplitude, f is the frequency and t is the time you would like to know the position of the mass. Not too bad, but you will want to be sure if you’re using this, you have switched the mode of your calculator from degrees to radians. More about what the heck radians are when we get to rotational motion. For our space-fairing astronaut, if their maximum displacement from equilibrium is 10 cm or 0.1 m, then at 5 seconds, he would be located at 3 cm or 0.03 m in the negative direction from the equilibrium position.
Graphs of position vs time can be used to determine what the velocity vs time graph looks like, and the velocity vs time graph can in turn be used to determine the shape of the acceleration vs time graph. Remember, slope of position time graphs is the velocity, and slope of velocity vs time is the acceleration. Have you forgotten? Go back and listen to episode 4. Using the law of conservation of energy, we can also find values for the velocity at any given moment if we know the spring constant, mass, and position of the oscillating object at the moment you want to know its velocity. Simply sum the elastic potential energy and the kinetic energy at any two points and solve for the velocity. You should see that the object has a maximum velocity and zero acceleration whenever it is at its equilibrium position. And it has a zero velocity with maximum acceleration whenever it is at a position equal to the amplitude. This location is sometimes referred to as a turning point. Wherever there is maximum velocity, there will be maximum kinetic energy, and whenever the object has reached its maximum displacement from the equilibrium position there will be maximum elastic potential energy.
For our 75 kg astronaut, we can find their maximum velocity achieved and maximum kinetic energy by setting the kinetic energy at the equilibrium position equal to the elastic potential energy at the point of maximum stretch (0.1 m). Knowing the spring constant of 350 N/m we can solve for the astronaut’s max velocity to be 0.216 m/s and their maximum kinetic energy would be 1.75 J.
To Recap…
The period of a spring is proportional to the square root of the mass and 1 over the square root of the spring constant. With the equations provided on the formula sheet, it is possible to find the position a mass is located at any given time as long as the amplitude and frequency are known. Finally, leveraging material from earlier units like kinematics or work and energy can help you find many other quantities for a mass on a spring. Physics works not just in specific cases, but universally. And your understanding of the world around you just keeps getting more and more clear.
Coming up next on the APsolute RecAP Physics 1 Edition, we look at rotational kinematics and let the work from unit one come around full circle.
Today’s Question of the Day focuses on spring oscillators.
Question:
How can the period of mass on a spring be doubled?