To Measure the Force of Limiting Friction for Rolling of a Roller on a Horizontal Plane
Objective
To determine the force of limiting friction when a roller rolls on a horizontal plane and to find the coefficient of rolling friction.
Theory
When a roller (such as a cylinder or a sphere) rolls on a horizontal plane, it experiences a resistive force known as rolling friction. Unlike sliding friction, rolling friction occurs due to deformation between the surfaces in contact.
The force of rolling friction \(F_r\) is given by:
$F_r = \mu_r \times N$
Where:
$\mu_r$ = coefficient of rolling friction
$N$ = normal reaction = $mg$
$m$ = mass of the roller
$g$ = acceleration due to gravity
For a roller on an inclined plane, the limiting condition occurs when:
$F_r = mg \sin \theta$
Where $\theta$ is the angle of inclination at which the roller just begins to move.
The coefficient of rolling friction is related to the critical angle by:
$\mu_r = \tan \theta$
Another way to express rolling friction is through the concept of the moment of the friction force:
The moment of rolling friction is given by:
$M = F_r \times R$
Where:
$M$ = moment of rolling friction
$R$ = radius of the roller
For a cylinder of mass $m$ and radius $R$ on a horizontal surface, the coefficient of rolling friction $\mu_r$ can also be calculated as:
$\mu_r = \frac{b}{R}$
Where $b$ is the displacement of the normal reaction from the center of the roller.
Apparatus Required
- A wooden board or horizontal plane
- A cylindrical roller (made of wood, metal, or other material)
- A pulley
- A pan with weights
- Slotted weights
- A measuring scale
- A thread or string
- A stopwatch
- A protractor (for measuring angles)
- A spirit level (to ensure the plane is horizontal)
Experimental Setup
Figure 1: Arrangement of roller, string, pulley, and weight pan to measure rolling friction
Experimental Procedure
- Place the wooden board on a table and ensure it is perfectly horizontal using the spirit level.
- Measure and record the mass ($m_1$) and radius ($R$) of the cylindrical roller.
- Arrange the experimental setup as shown in Figure 1, with the roller on the horizontal plane, a string attached to the roller passing over a frictionless pulley, and a weight pan attached to the other end of the string.
- Start with a small mass ($m_2$) in the weight pan.
- Observe whether the roller begins to roll. If not, gradually increase the mass in the weight pan until the roller just begins to roll with uniform velocity.
- Record this critical mass ($m_2$) at which the roller begins to roll uniformly.
- The force of limiting friction ($F_r$) is equal to the weight of the mass $m_2$: $F_r = m_2 \times g$
- Calculate the coefficient of rolling friction using: $\mu_r = \frac{F_r}{m_1 g} = \frac{m_2}{m_1}$
- Repeat the experiment for different masses of the roller and calculate the average coefficient of rolling friction.
Observations
Mass of the roller ($m_1$) = __________ kg
Radius of the roller ($R$) = __________ m
| S.No. | Mass in Pan ($m_2$) (kg) | Force of Limiting Friction $F_r = m_2g$ (N) | Normal Force $N = m_1g$ (N) | Coefficient of Rolling Friction $\mu_r = \frac{F_r}{N}$ |
|---|---|---|---|---|
| 1 | ||||
| 2 | ||||
| 3 | ||||
| 4 | ||||
| 5 |
Average coefficient of rolling friction ($\mu_r$) = __________
Alternative Method: Inclined Plane
Another method to determine the coefficient of rolling friction is using an inclined plane:
Figure 2: Arrangement of an inclined plane to measure rolling friction
- Place the roller on the inclined plane.
- Gradually increase the angle of inclination until the roller just begins to roll down with uniform velocity.
- Measure this critical angle $\theta$ using a protractor.
- Calculate the coefficient of rolling friction using: $\mu_r = \tan \theta$
- Repeat for different angles and calculate the average coefficient.
| S.No. | Critical Angle ($\theta$) (degrees) | $\tan \theta$ | Coefficient of Rolling Friction ($\mu_r$) |
|---|---|---|---|
| 1 | |||
| 2 | |||
| 3 |
Sources of Error and Precautions
- The surface of the roller and the plane may not be perfectly smooth.
- The horizontal plane may not be perfectly level.
- The pulley may have some friction that is not accounted for.
- The string may have non-negligible mass or may not be perfectly inextensible.
- Air resistance might affect the motion of the roller.
- The acceleration due to gravity ($g$) may vary slightly based on location.
- Ensure that the horizontal plane is perfectly level using a spirit level.
- Use a frictionless pulley or apply lubricant to minimize pulley friction.
- Use a thin, lightweight string to minimize its effect on the measurements.
- Ensure that the roller rolls without slipping.
- Perform multiple trials and take the average for more accurate results.
- Avoid any vibrations or disturbances during the experiment.
- When using the inclined plane method, ensure that the angle is increased very gradually to find the critical angle precisely.
Discussion and Conclusion
In this experiment, we determined the force of limiting friction for rolling and calculated the coefficient of rolling friction. The results demonstrate that rolling friction is much smaller than sliding friction for the same materials.
Rolling friction arises due to several factors:
- Deformation of surfaces: As the roller rolls, both the roller and the plane deform slightly at the point of contact, creating a resistive force.
- Micro-slipping: Despite appearing to roll perfectly, there may be microscopic slipping between the roller and the surface.
- Adhesion between surfaces: Molecular attractions between the roller and plane surfaces contribute to friction.
The coefficient of rolling friction ($\mu_r$) typically ranges from 0.001 to 0.02, depending on the materials involved. This is much smaller than the coefficient of sliding friction, which explains why rolling is more efficient than sliding.
The energy lost due to rolling friction per unit distance is given by:
$E = F_r \times d = \mu_r \times mg \times d$
Where $d$ is the distance rolled.
Understanding rolling friction is crucial for:
- Designing efficient wheel-based transportation systems
- Optimizing ball bearings and other rolling-element bearings
- Calculating energy losses in mechanical systems
- Improving the efficiency of vehicles and machinery
- Designing sports equipment like balls and wheels