8.19:
Rolling Resistance: Problem Solving
Consider a lawn roller of mass 100 kg and radius 25 cm. The roller arm extends at a thirty-degree angle to the horizontal axis.
Neglecting the friction developed at the axle, what force is required to move the roller at a constant speed, provided the coefficient of rolling resistance is 25 mm?
Assuming that the resultant driving force acting on the handle is applied along the arm, a free-body diagram of the roller indicating the weight, normal force, and driving force is drawn.
The driving force can be resolved into its horizontal and vertical components.
The system's geometry can be used to determine the angle between the normal force and the vertical axis.
The moment equilibrium condition is then applied to contact point O. By substituting the values of weight, components of the driving force and the corresponding perpendicular distances, estimates the driving force.
If a force with a magnitude much lesser than the driving force is applied, the roller does not move.
8.19:
Rolling Resistance: Problem Solving
Rolling resistance, also known as rolling friction, is the force that resists the motion of a rolling object, such as a wheel, tire, or ball, when it moves over a surface. It is caused by the deformation of the object and the surface in contact with each other, as well as other factors like internal friction, hysteresis, and energy losses within the materials. Rolling resistance opposes the object's motion, requiring additional energy to overcome it and maintain movement. In practical applications, minimizing rolling resistance can improve energy efficiency and reduce wear on both the rolling object and the surface.
Consider a lawn roller with a mass of 100 kg, a radius of 25 cm, and a coefficient of rolling resistance of 25 mm. The roller arm extends at a 30º angle to the horizontal axis. The force required to move the roller at a constant speed can be determined, provided the friction developed at the axle is neglected.
Assuming that the resultant driving force acting on the handle is applied along the arm, a free-body diagram of the roller can be drawn, including the weight (W), the normal force (N), and the driving force (F). The driving force can be resolved into its horizontal (Fh) and vertical (Fv) components.
Next, the system's geometry can be used to determine the angle θ1 between the normal force (N) and the vertical axis. The moment equilibrium condition is then applied to the contact point O.
The weight of the roller that equals the mass times the acceleration due to gravity is calculated as 98.1 N. By substituting the values of the weight, the horizontal and vertical components of the driving force, and the corresponding perpendicular distances, the driving force F is calculated as 120.86 N.
To maintain a constant speed, the horizontal component of the driving force must overcome the rolling resistance. If a force with a magnitude much lesser than the driving force is applied, the roller does not move. Understanding this relationship helps one appreciate the impact of different forces on the roller's motion and the importance of considering various factors when designing and using lawn rollers.
Suggested Reading
- Hibbeler, R.C. (2016). Engineering Mechanics ‒ Statics and Dynamics. Hoboken, New Jersey: Pearson Prentice Hall. pp 452 ‒ 453.
- Meriam, J.L.; Kraige, L.G. and Bolton, J.N. (2020). Engineering Mechanics ‒ Statics. Hoboken, New Jersey: John Wiley. pp 373.
- Beer, F.P.; Johnston, E.R.; Mazurek, D.F; Cromwell, P.J. and Self, B.P. (2019). Vector Mechanics for Engineers ‒ Statics and Dynamics. New York: McGraw-Hill. pp 463