6. Spring Balance The scale of a spring balance that reads...

6. Spring Balance The scale of a spring balance that reads from 0 to $15.0 \mathrm{~kg}$ is $12.0 \mathrm{~cm}$ long. A package suspended from the balance is found to oscillate vertically with a frequency of $2.00 \mathrm{~Hz} .$ (a) What is the spring constant? (b) How much does the package weigh?

Key Concepts

Weight (Force due to Gravity)

Weight is the force exerted on an object due to gravity and is calculated as the product of its mass and the acceleration due to gravity. In the context of a spring balance, knowing the weight of the suspended object allows for understanding how the force of gravity stretches the spring, which is linked to both equilibrium displacement and the oscillatory behavior in SHM.

Spring Constant

The spring constant is a parameter that quantifies the stiffness of a spring. It represents the ratio of the force exerted by the spring to its displacement from equilibrium. Determining the spring constant is crucial in problems involving oscillations and equilibrium conditions, as it directly influences the frequency and amplitude of the motion.

Simple Harmonic Motion

Simple Harmonic Motion (SHM) describes the oscillatory motion of systems like springs. In a mass-spring system, the natural frequency of oscillation is determined by the spring constant and the mass attached. The relationship f = (1/2?)?(k/m) allows one to connect the measurable frequency of oscillation to the spring's stiffness, enabling calculations of system properties such as the spring constant or the weight of the mass.

Hooke's Law

Hooke's Law explains that the force exerted by a spring is directly proportional to the displacement from its equilibrium position. This principle is foundational in understanding how spring balances work, as it relates the extension or compression of the spring to the applied force through the spring constant.

Transcript

00:01 So in this question we're asked to determine the spring constant of a weighing scale of spring and also the mass of a package, given the length of the spring used, the frequency of which a mass oscillates on it, and also the max and min values that the scale can weigh.

00:17 So for part a, we're asked to determine the spring constant.

00:22 So we know that the force exerted on the spring by the mass that's weighing, the magnitude of it f.

00:29 So the magnitude f is equal to the spring constant k times displacement x.

00:37 And we also know that the force that the mass would exert on the spring would be just its weight.

00:42 So that's equal to m g is equal to k over x.

00:47 So then we have an expression for k, so k is equal to m g over x.

00:57 And now we need to find a value of n sub into this and a value of x, such that we can since we know already what g is.

01:06 So we know that g is equal to 9 .81 meters per second squared.

01:12 So we need to sub in for all the other unknowns.

01:15 So since we're given that the maximum value that the spring can weigh is 15 kilograms, we're going to take that the maximum value of the mass coincides with the spring being compressed right down to its mass.

01:34 So the maximum compression of the spring would be 0 .12 meters, which is its length.

01:40 So we're going to assume that k is equal to, so the mass of max, m max, so the maximum mass times g over the length of the spring l.

01:51 So this is equal to 15 times 9 .81 over the length, maximum length, which is 0 .12 meters...

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