PRACTICAL WORK OF PHYSICS (MECHANICS)

We use the linear regression method only when the absolute uncertainty on \( x \) is \( ( \Delta x \approx 0 ) \) negligible  and the absolute uncertainty on  is constant  \( ( \Delta y = cste ) \) .

This method consists in drawing the best possible straight line of equation \( y=a+bx \)   which passes as close as possible to all the experimental points, as illustrated in figure 1.

\( b \) represents the slope of the regression line while  is the ordinate at the origin.

Knowing that we have \( n \)  measurements performed in practical work, then the values of \( a \)  and \( b \)  are given by the formulas:

The uncertainties associated with each parameter are given by:

If the linear regression line passes through the origin, we have the equation of the line \( y=bx \), in this case \(a=0\) and we obtain:

Experience 

A scale repairer wants to replace a defective spring in a scale. The spring must have a stiffness constant \( k=(3.00 \pm 0.05) N/m \) and a negligible mass. In his workshop, he found a spring of negligible mass but its stiffness constant is unknown.

Using Hook's Law :  \( F=k \times d \) , where \( F \) represents the force applied to the spring, \( k \)   the stiffness constant and \( d \) the elongation, he was able to calculate the value of \( k \) . As a result, he hung different masses on the spring and measured its elongation. Hook's law simplifies to : 

\( d= \frac{g}{k} m \)             (1)

The measurements are reported in the table below.

Questions

1. By comparing the physical equation (1) with the mathematical formula  \( y=bx \) , establish the following identifications : 
    \( x= .....................\), \( y= ..................... \) , \( b= ..................... \)   


2. Complete the table below :



i	\( m_i (kg) \)	\( d_i(m) \)	\( m_i d_i (…) \)	\( m_i^2 (…) \)	\( bm_i(…) \)	\( (d_i-bm_i )^2(…) \)
1	0.010	0.03290
2	0.020	0.06650
3	0.040	0.13280
4	0.060	0.19940
5	0.080	0.26590
\( n=... \)			\( \sum_{i=1}^{n} m_i d_i \) \(= ....................\)	\( \sum_{i=1}^{n} m_i^2 \) \(= ....................\)		\( \sum_{i=1}^{n} (d_i-bm_i )^2 \) \(= ....................\)




3. Give the numerical values of the following quantities with their corresponding units :
   \( b= .................... \) , \( \Delta d=.................... \) , \( \Delta b=.................... \) .


4. On the same graph sheet, represent the experimental points \( d=f(m) \), the error bars as well as the line of slope \(b\).


5. Calculate the spring stiffness constant and put it in the form \( k=(............. \pm .............) .... \).


6. Can the repairer replace the defective spring ? Explain. We give : \(g = 9.81 m/s^2 \).

1. Goals

• Highlight the movement of an elementary mechanical system: the oscillating pendulum.
• Determine the spring stiffness constant by two methods: static and dynamic.
• Measure the value of an unknown mass from the spring calibration curve.

2. Used Material

A support. A metal rod. A spring of negligible mass A box of marked masses. A graduated ruler. A stopwatch. A mass of unknown value.

3. Theory

3.1. Static study

The elongation of the spring is noted \(x_{eq} \) and is defined by: \(x_{eq} \ = \Delta l_0 = l - l_0\)

where:

• \( l_0 \) is the empty length of the spring
• \( l \) is the length of the extended spring.

A stiffness spring \( k \), whose mass will be neglected, is suspended vertically by its upper end from a support.

By applying Newton's first law, we have :

 At Equilibrium  :   \( \vec{P} + \vec{F}=\vec{0} \)  with  \( P = mg \)  and  \( F = k \Delta l_0 =k x_{eq} \)

From where, by projection on the axis of movement oriented vertically, we obtain:

\( P-F=0 \Rightarrow mg-k \Delta l_0 =0  \Rightarrow mg-k x_{eq} =0 \)

Thus, we have :

\( x_{eq}=\Delta l_0 = mg/k \)               (1)

3.2. Dynamic study

\(mg-k(\Delta x_{eq} + x ) = m \ddot{x} \)

4. Experimental Procedure

4.1. Static Study

1. Start by hanging the spring from the horizontal rod.
2. Attach the ruler so you can take precise measurements.
3. Measure the empty length \(l_0\) of the spring.
4. Then you must first suspend the weight rack to be able to place the masses on it.
5. Different known masses (\(m\)) of increasing values (see the table on the PW-sheet) are attached to the spring:
• At equilibrium, measure the corresponding elongations (\(X=x_{eq}\)), taking into account the mass of the weight rack.
• For each mass, take a minimum of three measurements (each student will take one measurement).
• Record your results in the table.
6. To preserve the spring, you must unhook the mass directly after performing the measurement. Never leave masses attached to the spring!!.
7. On a millimeter sheet, draw the calibration curve of the spring \(x_{eq}=f(m) \).

4.2.Determination of the Unknown Mass of a body

We want to determine the unknown mass \(m\) of a body from the calibration curve of the spring, for this :

1. Take the device used previously and put the unknown mass to the spring.
2. Measure the elongation \(x_{eq} \) of the spring.
3. Use the spring calibration curve to determine the value of the mass\(m \).

4.3.Dynamic Study

1. Resume the previous device. Attach a known mass \(m\) to the spring.
2. Stretch the spring (taking it away from its equilibrium position) a small distance \(x\), perfectly vertically, then let go of the mass without initial velocity.
3. Let the mass oscillate and measure the period of the oscillations \(T\) (read and carefully follow the measurement instructions on the sheet hung in the laboratory).
4. For each mass, take a minimum of three measurements (Each student will take one measurement).
5. Fill in the measurement table on the PW-sheet.
6. On a millimeter sheet, draw the calibration curve of the spring \(T^2=f(m) \)