To find the downward force, along an inclined plane, acting on a roller due to gravitational pull of the earth and study its relationship with the angle of inclination θ
1. Aim
To find the downward force along an inclined plane acting on a roller due to gravitational pull of the earth and to study its relationship with the angle of inclination θ by plotting a graph between force and sin θ.
2. Apparatus Used
- An inclined plane with protractor
- A roller of known mass
- Spring balance (or force sensor)
- Weight hanger and weights
- Thread or string
- Meter scale
- Graph paper
- Set square
- Clamp stand
- Spirit level
3. Diagram
4. Theory
When an object is placed on an inclined plane, it experiences a downward force due to the gravitational pull of the earth. This force can be resolved into two components:
- A force parallel to the inclined plane (causing the object to slide down)
- A force perpendicular to the inclined plane (the normal force)
If we consider a body of mass \(m\) placed on an inclined plane making an angle \(\theta\) with the horizontal, then:
The weight of the body, \(W = mg\) (acts vertically downward)
Component of weight parallel to the inclined plane = \(mg\sin\theta\)
Component of weight perpendicular to the inclined plane = \(mg\cos\theta\)
The component \(mg\sin\theta\) is responsible for the motion of the body down the inclined plane. As per Newton's Second Law of Motion, this force causes the acceleration of the body down the plane.
In this experiment, we will measure this downward force for different angles of inclination and verify that:
Force along the inclined plane, \(F = mg\sin\theta\)
By plotting a graph between the measured force \(F\) and \(\sin\theta\), we should get a straight line passing through the origin with slope equal to \(mg\). This will verify the relationship between the force and the angle of inclination.
5. Formula
Force along the inclined plane: \(F = mg\sin\theta\)
Where:
\(F\) = Force along the inclined plane (N)
\(m\) = Mass of the roller (kg)
\(g\) = Acceleration due to gravity (\(9.8\) m/s²)
\(\theta\) = Angle of inclination of the plane with the horizontal
From the graph of \(F\) vs \(\sin\theta\):
Slope of the graph = \(mg\)
Therefore, \(m = \frac{\text{Slope}}{g}\)
6. Procedure
- Set up the inclined plane on a firm horizontal surface. Use a spirit level to ensure the base is horizontal.
- Measure the mass of the roller using a balance and record it.
- Place the roller on the inclined plane and attach a string to it. Pass the string over a frictionless pulley at the end of the inclined plane.
- Connect the other end of the string to a spring balance or force sensor to measure the force required to prevent the roller from rolling down.
- Set the inclined plane at a small angle (around 10°) and measure the angle using the protractor attached to the plane.
- Note the reading of the spring balance or force sensor. This gives the force along the inclined plane.
- Increase the angle of inclination in steps (e.g., 15°, 20°, 25°, etc., up to about 45°) and repeat the force measurement for each angle.
- Record all observations in a tabular form.
- Calculate the value of \(\sin\theta\) for each angle.
- Plot a graph between the measured force (F) on the y-axis and \(\sin\theta\) on the x-axis.
- Determine the slope of the best-fit line through the origin.
- Compare the experimental value of \(mg\) (from the slope) with the theoretical value (\(m \times 9.8\)).
7. Observation Table
Given data:
- Mass of the roller (m) = _______ kg
- Acceleration due to gravity (g) = 9.8 m/s²
- Theoretical value of mg = _______ N
| S.No. | Angle of Inclination (θ°) | sin θ | Force along the inclined plane (F) in Newtons |
|---|---|---|---|
| 1 | |||
| 2 | |||
| 3 | |||
| 4 | |||
| 5 | |||
| 6 | |||
| 7 |
8. Calculations
Step 1: Calculate sin θ for each angle of inclination.
Step 2: Plot a graph with sin θ on the x-axis and force F on the y-axis.
Step 3: Draw the best-fit straight line through the origin.
Step 4: Calculate the slope of the graph:
Slope = \(\frac{\Delta F}{\Delta \sin\theta}\) = \(\frac{F_2 - F_1}{\sin\theta_2 - \sin\theta_1}\)
Step 5: Calculate the experimental value of mass:
Slope = mg
Therefore, m (experimental) = \(\frac{\text{Slope}}{g}\) = \(\frac{\text{Slope}}{9.8}\) kg
Step 6: Calculate the percentage error:
Percentage Error = \(\frac{|m_{theoretical} - m_{experimental}|}{m_{theoretical}} \times 100\%\)
9. Result
- The force acting on the roller along an inclined plane was measured for different angles of inclination.
- A graph was plotted between force (F) and sin θ, which showed a linear relationship passing through the origin.
- The slope of the graph was found to be _______ N.
- Mass of the roller (experimental) = _______ kg
- Mass of the roller (measured) = _______ kg
- Percentage error = _______ %
- The experiment verifies that the force acting on a body along an inclined plane is directly proportional to the sine of the angle of inclination, i.e., F = mg sin θ.
10. Precautions
- The inclined plane should be placed on a firm and level surface.
- The pulley should be frictionless to minimize errors.
- The angle of inclination should be measured accurately using the protractor.
- The spring balance/force sensor should be calibrated before use.
- The string should be inextensible and lightweight.
- The roller should be placed gently on the inclined plane to avoid any initial push.
- Ensure that there is no slipping between the roller and the inclined plane.
- Take multiple readings for each angle and use their average for better accuracy.
- Make sure the roller doesn't rotate during the experiment.
- Verify that the pulley is properly aligned with the inclined plane.
11. Sources of Error
- Friction between the roller and the inclined plane.
- Friction in the pulley.
- Error in measuring the angle of inclination.
- Error in the spring balance/force sensor reading.
- Non-uniform mass distribution in the roller.
- Slight deformation of the inclined plane under the weight of the roller.
- Air resistance acting on the moving parts.
- Stretching of the string under tension.
- Error in the measurement of the mass of the roller.
- Human errors in reading the instruments.
12. Viva Voice Questions
Q1. What is the principle behind this experiment?
A1. The principle is based on the resolution of forces. When an object is placed on an inclined plane, the gravitational force (weight) can be resolved into two components: one parallel to the inclined plane (causing the object to slide down) and one perpendicular to the plane (normal force). The component parallel to the plane is given by mg sin θ, which we verify in this experiment.
Q2. Why do we expect a linear relationship between the force and sin θ?
A2. According to the theory, the force along the inclined plane is given by F = mg sin θ. Since m and g are constants for a given roller and location, F is directly proportional to sin θ. This results in a linear relationship when F is plotted against sin θ, with slope equal to mg.
Q3. What would happen if the angle of inclination is 90°?
A3. At an angle of 90°, the inclined plane becomes vertical. In this case, sin 90° = 1, so the force along the inclined plane would be equal to mg, which is the entire weight of the roller. The roller would experience free fall if not supported.
Q4. Why is it important that the roller doesn't slip on the inclined plane?
A4. If the roller slips instead of rolling, additional frictional forces come into play, which are not accounted for in our theoretical model F = mg sin θ. Slipping would introduce errors in our measurements and invalidate our results.
Q5. How would the results change if the experiment was performed on the moon?
A5. On the moon, the acceleration due to gravity is approximately 1/6th that of the Earth. The formula F = mg sin θ would still hold, but the value of g would be different (about 1.62 m/s² instead of 9.8 m/s²). This would result in a smaller force for the same angle of inclination, and the slope of the F vs sin θ graph would be less steep.
Q6. What is the significance of the y-intercept in the F vs sin θ graph?
A6. Theoretically, the y-intercept should be zero because when sin θ = 0 (i.e., θ = 0°), the inclined plane is horizontal, and there should be no force acting along the plane. A non-zero y-intercept could indicate systematic errors in the experiment, such as friction or calibration issues.
Q7. How would the experiment change if you used a block instead of a roller?
A7. With a block, static and kinetic friction would play a significant role. Below a certain angle (determined by the coefficient of static friction), the block would not move. Above this angle, the force measured would be affected by kinetic friction. The relationship would still be linear with sin θ, but would include a friction term, and might not pass through the origin.
Q8. Why is it necessary to draw a best-fit line through the origin?
A8. Theoretically, the relationship F = mg sin θ suggests that when sin θ = 0, F should also be 0. By forcing the best-fit line through the origin, we are adhering to this physical constraint. It also helps in correctly determining the slope, which represents mg in our experiment.
Q9. What are the practical applications of understanding the relationship between force and the angle of inclination?
A9. This relationship is fundamental in many engineering applications such as designing ramps, roads on hills, roller coasters, conveyor belts, and calculating the forces in structures on slopes. It's also important in vehicle dynamics, civil engineering, and mechanical systems design.
Q10. How would you modify this experiment to measure the coefficient of friction between the roller and the plane?
A10. To measure the coefficient of friction, I would gradually increase the angle of the inclined plane until the roller just begins to move down the plane without any applied force. At this critical angle (θc), the component of weight parallel to the plane (mg sin θc) equals the maximum static friction force (μs × mg cos θc). By measuring this angle, I can calculate the coefficient of static friction using the formula μs = tan θc.