To determine the force constant of a helical spring by plotting a graph between load (force) and extension.

• A helical spring
• A spring support stand
• A set of standard masses (weights)
• A weight hanger
• A meter scale or vernier caliper
• Graph paper
• A pointer or marker

Figure 1: Experimental setup for determining the force constant of a helical spring

When an external force is applied to a spring, it gets stretched or compressed. According to Hooke's Law, the restoring force produced in the spring is directly proportional to the displacement from its equilibrium position, provided the elastic limit is not exceeded.

Mathematically, Hooke's Law can be expressed as:

Where:

• $F$ is the restoring force
• $k$ is the force constant (spring constant)
• $x$ is the displacement from the equilibrium position
• The negative sign indicates that the restoring force is in the opposite direction to the displacement

For our experiment, the external force is applied by hanging weights, and this force is balanced by the restoring force of the spring. Hence:

Where $m$ is the mass in kg, $g$ is the acceleration due to gravity (9.8 $m/s^2$), and $x$ is the extension in the spring.

According to this relationship, if we plot a graph of force $(F)$ versus extension $(x)$, we should get a straight line passing through the origin. The slope of this line gives the spring constant $(k)$.

The spring constant $(k)$ can be calculated using the following formula:

Where:

• $k$ = Force constant of the spring (N/m)
• $F$ = Force applied (N)
• $m$ = Mass (kg)
• $g$ = Acceleration due to gravity (9.8 $m/s^2$)
• $x$ = Extension in the spring (m)

From the graph of $F$ vs $x$, the spring constant is given by:

1. Set up the apparatus by fixing the spring to a rigid support.
2. Attach a pointer to the lower end of the spring to note the position clearly.
3. Record the initial position of the pointer (without any load) as the reference point.
4. Attach the weight hanger to the free end of the spring and note the new position of the pointer.
5. Calculate the extension produced by the weight hanger.
6. Now add weights in steps of 50g or 100g to the hanger and record the corresponding position of the pointer each time.
7. Calculate the extension for each load from the initial reference position.
8. Continue adding weights until you have at least 6-8 readings, but ensure the spring is not stretched beyond its elastic limit.
9. Record all observations in a tabular form.
10. Plot a graph of force (weight) versus extension.
11. Calculate the slope of the graph to determine the spring constant.
12. Repeat the experiment by gradually removing the weights and recording the corresponding positions to check for hysteresis effects.

S.No.	Mass (m) in kg	Force (F = mg) in N	Position of Pointer (cm)	Extension (x) in m	F/x (N/m)
1	0.050 (hanger)	0.490
2	0.100	0.980
3	0.150	1.470
4	0.200	1.960
5	0.250	2.450
6	0.300	2.940
7	0.350	3.430
8	0.400	3.920

Note:

• Initial position of pointer (without any load) = _________ cm
• Value of g taken = 9.8 $m/s^2$

From the observation table, we calculate:

1. Force (F) for each mass:

   $F = mg$

2. Extension (x) for each position:

   $x = \text{Current position} - \text{Initial position}$

   Note: Convert extensions from cm to m by dividing by 100.

3. Calculate the ratio F/x for each observation.

4. Plot a graph of Force (F) versus Extension (x) with:

   • Force on Y-axis
   • Extension on X-axis

5. Calculate the slope of the graph using:

   $\text{Slope} = \frac{\Delta F}{\Delta x} = \frac{F_2 - F_1}{x_2 - x_1}$

   Where $(F_1, x_1)$ and $(F_2, x_2)$ are two points on the straight line.

6. The slope of the graph gives the spring constant (k).

Calculation of Spring Constant:

The force constant of the given helical spring as determined from the experiment is:

The graph of force versus extension is a straight line, verifying Hooke's Law for the given spring within the experimental range.

1. The spring should be hung freely without any obstacles.
2. Weights should be added gently to avoid oscillations of the spring.
3. Readings should be taken only after the spring comes to rest.
4. The spring should not be stretched beyond its elastic limit.
5. The pointer should be clearly visible against the scale for accurate readings.
6. The base of the stand should be stable and on a horizontal surface.
7. Ensure that the initial reading (without load) is recorded accurately.
8. The weights used should be standard and calibrated.
9. The experiment should be performed at a constant temperature.
10. For better accuracy, repeat the experiment and take the mean value.

1. Friction in the experimental setup might affect the readings.
2. Non-uniformity in the spring coil can lead to uneven stretching.
3. The elastic limit of the spring might be exceeded without noticeable changes.
4. Temperature variations can affect the elasticity of the spring.
5. Inaccuracies in the measurement of extension.
6. Parallax error while reading the position of the pointer.
7. Initial stretching of the spring due to the weight hanger.
8. The spring may not strictly follow Hooke's Law over the entire range.
9. Variation in the value of acceleration due to gravity (g).
10. Oscillations in the spring while taking readings.