Conservation of Energy of a Ball Rolling Down an Inclined Plane
Using a Double Inclined Plane
Objective
To verify the law of conservation of energy by studying the motion of a ball rolling down on a double inclined plane.
Theory
The law of conservation of energy states that energy can neither be created nor destroyed but can be transformed from one form to another. The total energy of an isolated system remains constant over time.
When a ball rolls down an inclined plane, gravitational potential energy is converted into kinetic energy. For a pure rolling motion, the kinetic energy consists of both translational and rotational components.
Total energy at any point:
$E_{total} = E_{potential} + E_{kinetic}$
Gravitational potential energy:
$E_{potential} = mgh$
Kinetic energy (for a rolling sphere):
$E_{kinetic} = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 = \frac{1}{2}mv^2 + \frac{1}{2}(\frac{2}{5}mR^2)(\frac{v}{R})^2 = \frac{1}{2}mv^2(1 + \frac{2}{5}) = \frac{7}{10}mv^2$
For a solid sphere:
$I = \frac{2}{5}mR^2$ (moment of inertia)
$\omega = \frac{v}{R}$ (angular velocity for pure rolling)
Therefore:
$E_{kinetic} = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$
$= \frac{1}{2}mv^2 + \frac{1}{2}(\frac{2}{5}mR^2)(\frac{v}{R})^2$
$= \frac{1}{2}mv^2 + \frac{1}{5}mv^2$
$= \frac{7}{10}mv^2$
According to conservation of energy:
$mgh_1 = \frac{7}{10}mv_1^2 + mgh_2$
Where:
- $h_1$ is the initial height
- $h_2$ is the height at any point
- $v_1$ is the velocity at any point
For a double inclined plane, we can apply the conservation of energy between any two points. When a ball starts from rest at the top of the first incline (point A) with height $h_A$, rolls down and then up the second incline to point B with height $h_B$, we can write:
$E_A = E_B$
$mgh_A = mgh_B + \frac{7}{10}mv_B^2$
If the ball comes to rest at point B, then $v_B = 0$, and:
$mgh_A = mgh_B$
$h_A = h_B$
This means that in an ideal frictionless system, the ball should rise to the same height on the second incline as it started from on the first incline.
Apparatus Required
- A double inclined plane setup
- A metallic/wooden ball
- Meter scale
- Stopwatch
- Protractor
- Graph paper
- Spirit level
Experimental Setup
Fig 1: Double inclined plane setup with a ball at different positions showing energy conversion
Procedure
- Set up the double inclined plane on a horizontal surface. Use a spirit level to ensure the base is horizontal.
- Measure and record the angles of inclination ($\theta_1$ and $\theta_2$) of both planes using a protractor.
- Mark different positions (points) on the first inclined plane from which the ball will be released.
- Measure and record the vertical heights ($h_1$) of these points from the horizontal base.
- Release the ball from the first marked position without giving it any initial push.
- Observe and mark the maximum height ($h_2$) to which the ball rises on the second inclined plane.
- Repeat steps 5-6 for different release heights.
- For each trial, calculate the theoretical final height based on energy conservation principles.
- Compare the observed final heights with the theoretical predictions.
Observations
1. Angles of inclination:
- First inclined plane: $\theta_1 = \_\_\_\_\_$ degrees
- Second inclined plane: $\theta_2 = \_\_\_\_\_$ degrees
2. Data for different trials:
| Trial No. | Initial Height ($h_1$) in cm | Observed Final Height ($h_2$) in cm | Theoretical Final Height in cm | Percentage Error |
|---|---|---|---|---|
| 1 | ||||
| 2 | ||||
| 3 | ||||
| 4 | ||||
| 5 |
Calculations
1. For each trial, the theoretical final height can be calculated using:
$h_2 = h_1 - \text{energy loss due to friction}$
In an ideal system with no friction: $h_2 = h_1$
Example calculation for Trial 1:
If $h_1 = 30$ cm
Theoretical $h_2 = 30$ cm (assuming no energy loss)
If observed $h_2 = 27$ cm
Percentage error = $\frac{|30-27|}{30} \times 100\% = 10\%$
This error represents energy loss due to friction and other non-conservative forces.
2. Energy calculations at different points:
Initial potential energy: $E_{p1} = mgh_1$
Final potential energy: $E_{p2} = mgh_2$
Change in potential energy: $\Delta E_p = mg(h_1 - h_2)$
This change in potential energy equals the energy lost due to friction.
At the lowest point of the track (where the two inclines meet):
$E_{potential} = mg \times 0 = 0$ (assuming the reference level at this point)
$E_{kinetic} = \frac{7}{10}mv^2$
$v = \sqrt{\frac{10gh_1}{7}}$ (derived from energy conservation)
The kinetic energy at this point should equal the initial potential energy minus energy losses due to friction:
$E_{kinetic} = mgh_1 - \text{friction losses}$
3. Graph analysis:
Plot a graph with initial height ($h_1$) on x-axis and final height ($h_2$) on y-axis.
Slope of the graph represents efficiency of energy transfer.
Ideally, slope = 1 (perfect energy conservation)
Actual slope = $\frac{\Delta h_2}{\Delta h_1}$ (will be less than 1 due to energy losses)
Results and Conclusion
1. The graph between $h_1$ and $h_2$ gives a straight line with slope: _______
2. Percentage of energy conserved: $\frac{\text{Average observed } h_2}{\text{Average initial } h_1} \times 100\% = \_\_\_\_\_\%$
3. Average energy loss due to friction: $mg(\text{Average}(h_1 - h_2)) = \_\_\_\_\_$ J
The law of conservation of energy is verified if:
- The graph between $h_1$ and $h_2$ is approximately linear
- The energy loss calculated can be attributed to friction and air resistance
- The percentage of energy conserved is reasonably high (typically >85% for a well-set experiment)
Sources of error include:
- Friction between the ball and the inclined planes
- Air resistance
- Energy loss at the junction of the two planes
- Measurement errors in determining heights and angles
- Non-ideal rolling conditions (slipping)
Precautions and Sources of Error
- Ensure the base of the double inclined plane is perfectly horizontal using a spirit level.
- The ball should roll without slipping to maintain the relationship between translational and rotational kinetic energy.
- Release the ball gently without giving it any initial push.
- The junction between the two inclined planes should be smooth to minimize energy loss at that point.
- Take multiple readings for each height and use average values for calculations.
- Make sure the ball is perfectly spherical and uniform in density.
- Minimize air resistance effects by using an appropriate sized ball.
- Ensure accurate measurement of heights and angles.
Discussion Questions
- Why doesn't the ball reach exactly the same height on the second incline as it started from on the first incline?
- How would the results change if we used a hollow sphere instead of a solid sphere?
- What is the effect of changing the angle of the second inclined plane on the maximum height reached?
- How can we quantify the energy losses due to friction in this experiment?
- What modifications could be made to the experimental setup to minimize energy losses?