Spring Oscillations Lab Manual

To Study the Oscillations of a Mass in Combinations of Two Springs

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1. AIM

To study the oscillations of a mass when attached to:

  • A single spring
  • Two springs in series
  • Two springs in parallel

And to verify the relationship between the time period and effective spring constant for these combinations.

2. APPARATUS USED

  • Two helical springs of different spring constants (k₁ and k₂)
  • Slotted weights/mass set
  • Rigid support with clamp
  • Stop watch/timer
  • Meter scale/ruler
  • Weight hanger
  • Graph paper
  • Vernier caliper (to measure the diameter of the springs)

3. DIAGRAM

Oscillations of Mass with combination of two speings
Figure : Oscillations of Mass with combination of two springs

4. THEORY

When a mass is suspended from a spring and displaced from its equilibrium position, it exhibits simple harmonic motion. The motion is governed by Hooke's Law, which states that the restoring force is proportional to the displacement from equilibrium:

\[ F = -kx \]

Where:

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

For a mass suspended from a spring and undergoing simple harmonic motion, the time period (T) of oscillation is given by:

\[ T = 2\pi\sqrt{\frac{m}{k}} \]

Where:

  • \(T\) is the time period of oscillation
  • \(m\) is the mass attached to the spring
  • \(k\) is the spring constant

For Springs in Series:

When two springs with spring constants k₁ and k₂ are connected in series, the effective spring constant (k_series) is given by:

\[ \frac{1}{k_{series}} = \frac{1}{k_1} + \frac{1}{k_2} \]

Or:

\[ k_{series} = \frac{k_1 \times k_2}{k_1 + k_2} \]

For Springs in Parallel:

When two springs with spring constants k₁ and k₂ are connected in parallel, the effective spring constant (k_parallel) is given by:

\[ k_{parallel} = k_1 + k_2 \]

The time period for each arrangement can be calculated using the effective spring constant in the formula:

\[ T = 2\pi\sqrt{\frac{m}{k_{effective}}} \]

5. FORMULA

  1. Time period of oscillation:

    \[ T = 2\pi\sqrt{\frac{m}{k}} \]

  2. Effective spring constant for springs in series:

    \[ k_{series} = \frac{k_1 \times k_2}{k_1 + k_2} \]

  3. Effective spring constant for springs in parallel:

    \[ k_{parallel} = k_1 + k_2 \]

  4. Spring constant calculation from time period:

    \[ k = \frac{4\pi^2m}{T^2} \]

6. PROCEDURE

A. Determination of Individual Spring Constants (k₁ and k₂)

  1. Set up the experimental arrangement with a rigid support and clamp.
  2. Hang the first spring (Spring 1) from the support and attach a weight hanger to it.
  3. Measure the initial length of the spring without any additional weights.
  4. Add known weights (m₁) to the hanger and measure the extension of the spring.
  5. Calculate the spring constant k₁ using the formula k₁ = mg/x, where x is the extension produced.
  6. Repeat steps 3-5 for different masses and take the average value of k₁.
  7. Repeat the entire procedure for the second spring (Spring 2) to determine k₂.

B. Single Spring Oscillation

  1. Attach Spring 1 to the rigid support and hang a known mass (m) on it.
  2. Allow the system to come to equilibrium.
  3. Gently pull the mass downward by a small distance (2-3 cm) and release it to start the oscillations.
  4. Using a stopwatch, measure the time for 20 complete oscillations.
  5. Calculate the time period T = (Time for 20 oscillations)/20.
  6. Repeat steps 3-5 for different masses.
  7. Repeat the entire procedure for Spring 2.

C. Springs in Series

  1. Attach Spring 1 to the rigid support and connect Spring 2 below it.
  2. Attach a known mass (m) to the lower end of Spring 2.
  3. Allow the system to come to equilibrium.
  4. Gently pull the mass downward by a small distance and release it to start the oscillations.
  5. Measure the time for 20 complete oscillations and calculate the time period.
  6. Repeat for different masses.

D. Springs in Parallel

  1. Attach both springs side by side to the rigid support.
  2. Connect their lower ends to a straight rod or bar.
  3. Hang a known mass (m) at the center of the bar.
  4. Allow the system to come to equilibrium.
  5. Gently pull the mass downward by a small distance and release it to start the oscillations.
  6. Measure the time for 20 complete oscillations and calculate the time period.
  7. Repeat for different masses.

7. OBSERVATION TABLES

Table 1: Determination of Spring Constant k₁ (Spring 1)

S.No. Mass (m) kg Weight (mg) N Extension (x) m Spring Constant k₁ = mg/x (N/m)
1
2
3
4
5

Average value of k₁ = ________ N/m

Table 2: Determination of Spring Constant k₂ (Spring 2)

S.No. Mass (m) kg Weight (mg) N Extension (x) m Spring Constant k₂ = mg/x (N/m)
1
2
3
4
5

Average value of k₂ = ________ N/m

Table 3: Time Period of Oscillation for Single Spring (Spring 1)

S.No. Mass (m) kg Time for 20 oscillations (s) Time period T = (Time/20) s T² (s²) 4π²m (kg) k = 4π²m/T² (N/m)
1
2
3
4

Table 4: Time Period of Oscillation for Single Spring (Spring 2)

S.No. Mass (m) kg Time for 20 oscillations (s) Time period T = (Time/20) s T² (s²) 4π²m (kg) k = 4π²m/T² (N/m)
1
2
3
4

Table 5: Time Period of Oscillation for Springs in Series

S.No. Mass (m) kg Time for 20 oscillations (s) Time period T = (Time/20) s T² (s²) Theoretical k_series = (k₁×k₂)/(k₁+k₂) (N/m) Experimental k_series = 4π²m/T² (N/m)
1
2
3
4

Table 6: Time Period of Oscillation for Springs in Parallel

S.No. Mass (m) kg Time for 20 oscillations (s) Time period T = (Time/20) s T² (s²) Theoretical k_parallel = k₁+k₂ (N/m) Experimental k_parallel = 4π²m/T² (N/m)
1
2
3
4

8. CALCULATIONS

  1. For individual spring constants:

    \[ k_1 = \frac{mg}{x} \text{ (for Spring 1)} \]

    \[ k_2 = \frac{mg}{x} \text{ (for Spring 2)} \]

  2. For time period calculation:

    \[ T = \frac{\text{Time for 20 oscillations}}{20} \]

  3. For experimental spring constant from time period:

    \[ k = \frac{4\pi^2m}{T^2} \]

  4. Theoretical values:

    For springs in series: \[ k_{series} = \frac{k_1 \times k_2}{k_1 + k_2} \]

    For springs in parallel: \[ k_{parallel} = k_1 + k_2 \]

  5. Percentage error calculation:

    \[ \% \text{ Error} = \frac{|\text{Theoretical value} - \text{Experimental value}|}{\text{Theoretical value}} \times 100 \]

9. GRAPH

  1. Plot a graph of T² vs. m for each arrangement (single spring, series, and parallel).
  2. The slope of each graph should be 4π²/k, where k is the effective spring constant.
  3. Compare the slopes obtained from the graphs with the theoretical calculations.
Sample Graph of T² vs. m

10. RESULT

  1. The experimentally determined spring constants are:

    • k₁ = ________ N/m (Spring 1)
    • k
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