Question

You have a linear (following Hooke's Law) spring with an unknown spring constant, a standard mass, and a timer. Explain carefully how you could most practically use these to measure masses in the absence of gravity. Be as quantitative as you can. Regard the mass of the spring as negligible.

Short answer

Question: Explain how to use a linear spring, a standard mass, and a timer to measure masses in the absence of gravity. Answer: To measure masses in the absence of gravity, we can rely on the simple harmonic motion (SHM) of a spring-mass system. First, attach the standard mass to the spring, displace it slightly from its equilibrium position, and measure the period of oscillation using the timer. Calculate the spring constant (k) using the known mass and period. Next, attach the unknown mass to the spring and measure its period of oscillation. Using the calculated spring constant and the period of oscillation for the unknown mass, you can calculate the mass of the unknown object using the equation m2 = (k*(T2/(2*pi))^2), where T2 is the period of oscillation of the unknown mass.

Step-by-step solution

Understand the given system and Hooke's Law

We have a spring that obeys Hooke's Law, which means the force exerted by the spring is proportional to its displacement from its equilibrium position. Mathematically, this can be represented as F = -kx, where F is the force exerted by the spring, k is the spring constant, and x is the displacement of the spring from its equilibrium position. Since there is no gravity, we can assume that the spring will only be stretched or compressed by the mass attached to it.

Identify the connection between the mass and the spring

In this scenario, one way to connect the mass and spring constant is by considering the simple harmonic motion of the spring-mass system. When a mass is attached to a spring and displaced from the equilibrium position, it will oscillate back and forth with a certain period which only depends on the mass and the spring constant. This period, represented by T, can be found using the equation: T = 2*pi*sqrt(m/k) Where T is the time period of oscillation, m is the mass attached to the spring, k is the spring constant, and pi is a constant approximately equal to 3.14159.

Calculate the spring constant using the standard mass

Attach the standard mass (let's call it m1) to the spring and displace it slightly from its equilibrium position. Start the timer and count the number of oscillations (n1) for a duration (t1). Then, using the equation T1 = t1/n1, calculate the period T1 for the standard mass m1. Using the period equation, we can now find the spring constant: k = (m1*(2*pi/T1)^2)

Measuring unknown mass using the spring constant

Now that we have calculated the spring constant (k), we can use it to measure any other unknown mass (m2). Attach the unknown mass to the spring, displace it from its equilibrium position, and start the timer. Count the number of oscillations (n2) for a duration (t2). Calculate the period T2 for the unknown mass m2 using the equation T2 = t2/n2.

Calculate the unknown mass

Finally, we can calculate the unknown mass (m2) by using the period of oscillation and the spring constant, using the period equation mentioned in step 2: m2 = (k*(T2/(2*pi))^2) The calculated value of m2 is the unknown mass in the absence of gravity. This method is practical because it only relies on the properties of the spring, the mass, and the timing of oscillations, excluding the gravitational force.

Key concepts

Hooke's Law

Imagine stretching a spring attached to a hook on your wall. As you pull, you'll feel a resistance, an urging of the spring to return to its original shape. This is where Hooke's Law comes into play. It's a principle stating that the force needed to extend or compress a spring is directly proportional to the distance it is stretched or compressed. Now, without gravity, we're focusing solely on this fascinating tug-of-war between force and motion.

In mathematical terms, this law is epitomized by the equation \( F = -kx \), where \( F \) is the force exerted by the spring, \( k \) is the spring constant, and \( x \) is the spring's displacement from its natural, unstressed position. Thinking of it like an invisible rubber band, the longer you stretch (or compress), the greater the force that springs back. This simple concept becomes the cornerstone for measuring mass with a spring, as the reaction of the spring—the force it produces—is a reliable indicator of the mass that burdens it.

Simple Harmonic Motion

Ever watched a playground swing? It moves back and forth in a predictable, flowing manner. This is a showcase of simple harmonic motion—a movement that involves a restoring force proportional to the displacement and that acts in the opposite direction. When we deal with a spring and mass system, without the complication of gravity, it's a classic example of simple harmonic motion.

After stretching or compressing the spring with a mass and letting go, you'll observe it performing a dance, oscillating to and fro. This rhythmic motion is both steady and systematic, defined by the mass at one end and the spring constant at the other. No matter how much it moves, the system will always pass back through the equilibrium position eventually, like a pendulum swinging with unerring regularity. It's this unchanging motion we exploit to measure unknown masses.

Oscillation Period

Timekeeping is essential in our lives, and it doesn't stop short when it comes to physics experiments. The oscillation period is the time it takes for one complete cycle of motion, like one full swing of that playground swing we imagined before. For a mass on a spring, the period \( T \) of oscillation is a window into the heart of the system's dynamic.

To capture the essence of an oscillation period, we use the formula \( T = 2\pi\sqrt{m/k} \)—a crystal-clear expression relating mass, spring constant, and time. With \( m \) representing the mass and \( k \) the spring constant, the equation reveals that the period depends squarely on these two variables. So, by timing the swings or oscillations of the spring system, you're essentially decoding its mass properties, turning time into a scale to weigh masses—truly a temporal tape measure.

Spring Constant Calculation

The spring constant, symbolized by \( k \), is the rigidity superstar of the spring world. It tells us how stiff or lax the spring is. Think of \( k \) as the spring's character trait, describing how much it resists being stretched or squished. Calculating this trait allows us to predict the spring's reaction to a variety of masses.

Let's talk practicals. Say you've got a spring and a mass that you know, a standard for comparison—your benchmark. You jiggle it to find out its oscillation period using \( T = t/n \) where \( t \) is the total time you've observed the oscillations and \( n \) is the number of cycles. With that period \( T \) in hand, bingo, you invert the formula we talked about earlier to solve for \( k \)—\[ k = (m\cdot(2\pi/T)^2) \]. Now you've got a key to unlock the mass of any other mysterious object. Affix it to the spring, and repeat the dance: measure the time, count the oscillations, calculate the period, and you're all set to solve for the unknown mass using the relationship we've unraveled between time and mass.