TP PHYSIQUE 1 (Pratical Work in Mechanics)

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 : \(g = 9.81 m/s^2 \).