Motion of Hollow Cylinders

This extended experimental investigation was performed in order to examine how the physics principles apply to the rolling motion of wheels and other objects down an incline and how these findings can relate to the design of skateboard wheels. More specifically, to test the principle of conservation of energy as it applies to the motion of rolling objects, in addition to the factors that affect this rolling motion.

It was discovered during pre-experimental research that the wheels mass, radii and length will have an effect on the velocity at which it rolls down an incline. The first set of experiments was devoted to testing the effect of these factors individually to clarify or disprove this claim. The claim was discovered to be correct, which allowed for further testing to be commenced to determine exactly what factors did affect its velocity.

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The next experiment involved changing the structure of the wheel to see its outcome, by racing thin-walled and solid wheels, as well as wheels with different cross-sectional areas. Because it was found previously that mass, radii, and length had insignificant effect on the velocity at which the objects came down, it was not important to keep the dimensions and the masses of the objects the same. All the objects dealt with thus far all had their centre of gravity through their centre. Thus, the next experiment changed the position of the centre of gravity of a wheel to see its consequence. Furthermore, the effect of having variable amounts of fluid inside a can as it rolls down the incline was investigated. Lastly, the angle of inclination was changed to determine its relationship to the velocity at which the objects roll down.

The results of this investigation reinforced the fact that the final velocity of the object is independent on its mass, radii and length. The objects shape and the height at which it was released were found to have an effect on its final velocity. It was postulated that the water inside a can as it rolls down the incline, does not roll with the can, but slides, reducing the objects rotational inertia. However, throughout the course of this investigation, certain errors in the design of the experiment and the procedure would have had an effect upon the accuracy of the results.
To make this investigation simpler, it was assumed that the objects rolled without slipping. Before we can continue explaining the background theory to this investigation, it must be understood what is meant by rolling without slipping.

Rolling without slipping depends on the static friction between the rolling object and the ground. Static friction keeps an object from moving when a force is applied (Cutnell, p.96). For example, a person trying to push a heavy object is unable to budge it because of static friction. Static friction depends on the normal force exerted on the object, which is equal to its weight if the object is resting on a level surface. It is also dependent on the relative roughness of the surface. Surfaces when examined microscopically show irregular features that consist of numerous bumps and depressions (see below). This contouring causes surfaces to interlock and impedes movement. The coefficient of static friction, is a measure of the relative roughness between the surfaces (Myers, 2006; p.46).

Surfaces, even though they may appear and feel smooth, are highly irregular, giving rise to frictional forces. An object resting on an incline begins to slip when the force, that is , is larger than the static frictional force , whereis the static friction coefficient. Hence, as the inclined plane is elevated, the rolling object will also slip. However, by maintaining a small angle of elevation (?<25�) the amount of slip will be irrelevant. Likewise, by increasing the static friction coefficient the possibility of the object slipping reduces.

When an object does not slip, its point of contact has zero velocity with respect to the surface (as seen in the below figure of a bicycle wheel). Note that if the point of contact does not have zero velocity than slipping occurs. The bicycle wheel of radius R, will only roll without slipping if the circular arc, s, is equals the distance, d, through which an axle moves. When it doesn’t slip, the contact point will have zero velocity. Image adapted from Cutnell, 2004; 218.

Notice that if the object does not slip, then the circular arc length s is equal to the distance d (Cutnell, 2004; 218-219). By dividing both sides by the time t it takes for B to be in contact with the ground, we derive: The term d/t is equal to the linear speed v as it represents the speed at which the axle moves parallel to the ground. The term s/t represents the tangential speed and is related to the angular speed about the axle according to: tangential speed = R.

Therefore, the linear velocity, v, of the centre of the wheel and the angular speed, ? (in rad/s), of the wheel about its centre are related by: (Equation 1) Because we are dealing with a rigid object, all lines in it (i.e. the spokes in the bicycle wheel) rotate through the same angle at the same time. Thus, at any given instant, every part of the object has the same angular velocity (Giancoli, 2005; 196).

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