Motion of Hollow Cylinders

PHYSICS: EXTENDED EXPERIMENTAL INVESTIGATION

∙ THE DESIGN OF A SKATEBOARD WHEEL ∙

PART I – EXPERIMENTAL RESEARCH

ABSTRACT

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. 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.

INTRODUCTION

Rotation about a moving axis – rolling motion

Figure 1 – The moon rotates around its axis while it orbits the earth. Thus, it is an example of a body that rotates about a moving axis.

Image courtesy of: Whitman, Moon motion and tides, (Internet).

There are three basic types of motion of a rigid body (an object whose size and shape does not change as it moves):

translation motion in which the object as a whole moves along a trajectory but does not rotate,
rotational motion where the object rotates about a fixed point, and
combination motion where an object rotates along a trajectory (as illustrated in Figure 1).

Rolling is a combination motion in which an object rotates about an axis that is undergoing translational motion (Knight, 2004; p.393). This investigation is based around the concept of rolling motion down an inclined plane, and hence an understanding of both translation and rotational motion is important.

Rolling without slipping

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 Figure 2 below). This contouring causes surfaces to interlock and impedes movement. The coefficient of static friction (µ), is a measure of the relative roughness between the surfaces (Myers, 2006; p.46).

Figure 2 – 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 (θ<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.

Figure 3 – 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 ω about the axle according to: tangential speed = Rω. Therefore, the linear velocity, v, of the centre of the wheel and the angular speed, ω (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).

The same kind of reasoning used above reveals that linear and tangential acceleration, a and α (in rad/s2), are related by:

(Equation 2)

Note that these relationships are only valid for when an object rolls without slipping.

Relations between angular and linear quantities

In the above section, we showed that angular velocity, ω, is defined in analogy with linear velocity, and also for angular acceleration, α, with translational acceleration. The angle an object has rotated from a reference line is similarly related to linear displacement x. The symbols of linear and rotational kinematics are thus summarised as:

Note: The angle is measured in radians. Also, for the vector rotational motion quantities, direction can be specified with a + when rotating counter clockwise, or a – sign when rotating clockwise (Serway et al, 2006;193).

These rotational motion symbols can be substituted in for there corresponding linear ones to derive equations for rotational kinematics under constant angular acceleration.

Table 2 adapted from (Tipler, 1999; 274) & (Young et al, 2007; 272).

Torque

We have thus far discussed rotational kinematics by describing rotational motion in terms of angle, angular velocity and angular acceleration. In this investigation however, it is important to understand the dynamics, or cause, of rotational motion.

In translational motion we have seen how a net external force causes the object to accelerate. However, as discussed above, a rigid body can also have angular acceleration. So what causes its rotational motion to change?

According to Randall Knight (2004; 378), a change in rotational velocity results from a net external torque instead of a net external force. Like force, torque has a sign. A positive torque causes a rigid object to rotate in a counter-clockwise direction, whereas a negative torque causes the object to rotate clockwise. However, it must be noted that torque differs from force in that it is measured about a pivot point. To consider the effect of the pivot point and other factors that effect torque, consider the commonplace situation of pushing open a door.

Figure 4 – The door handle is situated where it is, because by applying a force F2, it is the most effective way of opening the door.

As can be seen in Figure 4, five pushing forces of equal magnitude are shown. From experience, it is known that forces F1 and F5 will not open the door. The force F2 will be the most effective, while the force F4 (in the middle of the door) will be half as effective as F2. F3 having an angle of 45° will open the door but be less effective than F2. It can also be noted that if the magnitude of the forces are reduced by half, they will be half as effective at opening the door. The reasons for these observations can be clarified by the definition of torque below.

From these findings it can be concluded that the ability to cause rotation (torque) of the door is dependent on the magnitude of the force, the distance from the point of application from the hinge and on the direction the force is applied. The torque, τ, can hence be defined as:

Torque = Magnitude of the force x Lever arm

τ = fR sin Ø (Equation 7)

where the lever arm is the distance R between

the line of action and the axis of rotation (as

shown if Figure 5 to the right), and where Ø

is the angle at which the force is applied. The

SI unit of torque is Newton meter (Nm)

(Cutnell,2004;232).

Rotational dynamics – Newton’s second law

In the above section we determined that torque is the rotational equivalent of force. From the observations of the swinging door it is clear that acceleration is proportional to the net applied torque. This corresponds to Newton’s second law for translational motion where linear acceleration is proportional to the net force. Notably, it is also inversely proportional to the inertia of the object (Knight, 2004; 381).

Because all the points in a rotating object have the same angular acceleration, it will be useful to express Newton’s second law in terms of angular acceleration. From Equation 2 we know that tangential and angular accelerations are related by . Newton’s second law can now be written as:

(Equation 8)

Multiplying both sides by R gives us:

(Equation 9)

However, this applies to only a single particle. For an object with many particles which have the same angular acceleration, the net torque is given by:

(Equation 10)

The quantity in Equation 10 above is the proportionality constant between angular acceleration and net torque, is termed the moment of inertia I (in kgm2). Substituting I into Equation 9 best conveys the idea that torque is the cause of angular acceleration.

(Equation 11)

Rotational dynamics – Moment of inertia

For this investigation, a better understanding of rotational inertia is important. Firstly, from Equation 11 above, it is clear the moment of inertia is the rotational equivalent of mass. The moment of inertia has the same function in Equation 11 as it does as the inertial mass (m) has in F= ma. Objects with larger mass have a larger inertia, and thus are harder to accelerate (Knight, 2004; 382). Similarly, an object with a larger moment of inertia is harder to rotate.

From Table 3 below, it is clear that the moment of inertia is dependent on the mass and the radius on the included objects, but it is also dependent on the distribution of that mass. Consider the following two bicycle wheels:

Figure 6 – Two differently weighted bicycle wheels. The shaded area represents the majority of the mass.

The above objects have the same total mass and same radius. Having the mass near the centre reduces the moment of inertia making it easier to spin. Thus, cyclists use the lightest possible tires, which have the majority of their mass concentrated around the centre.

Table adapted from (Tipler, 1999; 264).

Tying the above concepts back to rolling objects down an inclined plane

Consider the following cylinder rolling down an inclination:

Figure 7 – A cylinder with radius R rolling down an inclined plane.

It must be noted that the cylinder in Figure 7, or any object for that matter, will not roll down the incline without friction. This is because the torque needed to allow angular acceleration can only be supplied by the frictional force. The other two forces act through the centre of the rolling object and hence produce zero torque.

Because the angle of inclination is small, it can be assumed that the cylinder will be rolling without slipping. Therefore, the bottom of the wheel is going to be momentarily at rest, forbidding friction from dissipating energy. The frictional force is doing no work (Rolling motion, Internet).

The rolling cylinder will have an angular acceleration that satisfies Equation 11 above – that is, . In this case, the force is the frictional force f, and R is the radius of the cylinder. The frictional force can then be isolated to give:

(Equation 12)

Recalling the Equation 2 gave us the angular acceleration in terms of translational acceleration, we can substitute the result into Equation 12 to give:

(Equation 13)

By applying Newton’s second law, the linear acceleration directed downward along the track can be determined:

(Equation 14)

By substituting in the result in Equation 13 into Equation 14, we derive:

(Equation 15)

It must be noted that the above equations only apply for when the object rolls without slipping. It is obvious from Equation 15 that the characteristics of the rolling object only come into play in the ratio . For a solid cylinder whose moment of inertia is ½mR2 (Table 3), its linear acceleration is 2/3 gsinθ ms-2. The below table summarises this acceleration with other shapes.

From the above table it is clear that the solid sphere will have the greatest acceleration, then the solid cylinder, then the thin spherical shell, and lastly the cylindrical shell.

Conservation of mechanical energy in rolling motion down an incline

As stated in the above section Rolling without slipping, an object will slip without rolling down an incline if there is no friction, as it is friction that provides the torque. When the object does not slip, the point of contact between the rolling object and the surface is instantaneously at rest, and thus, the frictional force will do no work. It is gravity alone that does work on the object, and hence energy is conserved, with the sum of the kinetic energy and gravitational potential energy remaining constant throughout the objects descent.

Firstly consider the object:

Figure 8 – A rigid object moving forward while it rotates around an axis going through the centre of mass

The object in Figure 8 is moving forwards with velocity v. Hence it has translational kinetic energy:

(Equation 16)

But the object is also rotating around the axis with an angular velocity ω. Thus it also has rotational kinetic energy:

(Equation 17)

If the object is placed at the top of an incline and is held in this position, all of its energy will be in the form of gravitational potential energy:

(Equation 18)

where h is the height at which the object is raised.

The total energy of the object can hence be written as (Rolling Motion, Internet):

(Equation 19)

When the object at the top of the incline begins to roll, it will loose potential energy and gain kinetic energy. If ΔKE is the difference in kinetic energy at two points along the incline, and ΔPE is the difference in potential energy at those same two points, then according the conservation of energy (Rolling Motion, Internet):

(Equation 20)

At the end of the incline, all potential energy will have been transferred into kinetic energy. Hence:

(Equation 21)

Substituting the result from Equation 1, we get:

(Equation 22)

Similarly to the section above, we can determine the theoretical final velocities of the shapes used in this experiment:

It can be noted that the larger the term in from of the mR2 value the slower the objects theoretical final velocity. It can also be seen, that the mass, radii and length of the objects will have in theory, no effect on the final velocity at which the objects come down. It is only the moment of inertia that has an effect.