Question

LAB PROBLEM 1: MEASURING SPRING CONSTANTS TWO WAYS You are selecting springs for a large antique clock; to determine the forces they will exert in the clock, you need to know their spring constants. One book recommends a static approach: hang objects of different weights on the spring and measure the displacement from equilibrium. Another book suggests a dynamic approach: hang an object on the end of a spring and measure its oscillation frequency. You decide to compare the results of the two methods, in order to get the best precision possible for your characterization of the clock s springs. But first you have to figure out how to calculate spring constants from each type of measurement. Instructions: Before lab, read the laboratory in its entirety as well as the required reading in the textbook. In your lab notebook, respond to the warm up questions and derive a specific prediction for the outcome of the lab. During ab compare your warm up responses and prediction in your group. Then, work through the exploration, measurement, analysis, and conclusion sections in sequence, keeping a record of your findings in your lab notebook. It is often useful to use Excel to perform data analysis, rather than doing it by hand. Read: Mazur Chapter 8, especially Sections 8.6 and 8.9, and Sections 15.1-15.6. EQUIPMENT You have s a table clamp, rod, meterstick, stopwatch, mass set and the video analysis equipment. You can hang the spring from a rod that is extended from a table clamp

Can you help me to solve these questions basically?

Answer 1

When the spring is in unstretched position then restoring force acting on mass (Say m) due to spring will be exactly balanced by the force due to gravity on the mass.

Let its equilibrium position be y_o. then using balance of forces,

  

  

where k= spring constant

g= acceleration due to gravity

or $y_o= -\frac{mg}{k}$ ................................(1)

Now let mass be stretches to a new position, say, y_n. Then it will have net acceleration in upward direction due to more spring restoring force acting on it. Thus we'll have

(Using balance of forces and Newton's Laws of motion)

(Taking upward direction as positive)

$-k(y_n+\frac{mg}{k})= ma$

using equation 1 in above,

let y_n-y_o= y = net displacement from its equilibrium position.

also acceleration in vertical direction can be written as $a= \frac{d^2y}{dt^2}$

then we'll have

   $m\frac{d^2y}{dt^2}= -ky$

$\frac{d^2y}{dt^2}+\frac{k}{m}y=0$ ............................(1)

Let the solution of above equation be $y= Asin\omega t$ .....................(2) ( where A= some constant)

then $\frac{dy}{dt}= -A\omega cos\omega t$

and $\frac{d^2y}{dt^2}= -A\omega^2 sin\omega t= -\omega^2 y$ .........................(3)

Putting equation 2 and 3 in equation 1, we get

$-\omega^2 y+\frac{k}{m}y=0$

$-\omega^2 +\frac{k}{m}=0$

we'll get angular frequency of this oscillation as

   $\omega= \sqrt{\frac{k}{m}}$ (ANS)

And if the Time Periof of oscillation is "T" then angular frequency in terms of Time Period is written as

$\omega= \frac{2\pi }{T}$ (ANS)