ON THE VELOCITY OF A WEIGHTED CYLINDER DOWN AN INCLINED PLANE

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1 ON THE VELOCITY OF A WEIGHTED CYLINDER DOWN AN INCLINED PLANE Raghav Grover and Aneesh Agarwal RG (Grade 12 High School), AA (Grade 11 High School) Department of Physics, The Doon School, Dehradun. raghav @doonschool.com, aneesh @doonschool.com Abstract: A hollow cylinder consisting of a PVC pipe with a straw filled with sand stuck along its length on its inner surface, causing uneven distribution of mass, was rolled down an inclined plane surface. The video of the motion was captured using a slow motion camera and analysed using (Open Source Physics) Tracker software. The geometric centre of the weighted cylinder was tracked and the velocity-time graph was modelled using the equation of a cycloid. 1. INTRODUCTION The physics of rolling is thought to be completely understood in terms of classical mechanics. The motion of rolling down an inclined plane is explained using the simple concepts of angular velocity and acceleration due to gravity. Our experiment was designed such that the motion of a hollow cylindrical PVC pipe rolling down an inclined plane is modified by sticking a straw filled with sand on the inner surface of the hollow cylinder along its length, creating a weighted cylinder. In our primary hypothesis it was assumed that the geometric centre of the weighted cylinder will follow a sinusoidal trajectory along the parabolic curve arising from the accelerated motion down the inclined plane (in a displacement-time graph). Based on this hypothesis, experimental trials were conducted using a slow-motion video camera and analysed using Open Source Physics based TRACKER software. Upon investigation of the experimental data it was found that the sinusoidal hypothesis was incorrect, as the velocity-time graph was a reflected compressing cycloid along a linear slope. This velocity-time graph was modelled using the cartesian equation of a cycloid on a linear slope. 2. METHOD AND MATERIALS An MDF hardboard was procured and cut into an arbitrary size of 4 ft by 1.5 ft. This hardboard was reinforced by wooden sticks and covered with cotton cloth in order to ensure a smooth, flat surface. This board was tilted at an approximate angle of 5. A fixed release position was marked on the board. A white chart paper was placed in the background to facilitate easy tracking in TRACKER software. The scene was provided natural and artificial lighting both, such that the slow motion video is clear enough for tracking. For preparing the cylinder, a 2.5 inch diameter PVC pipe was cut at a length of one straw. A masking tape cross was made on one end of the cylinder, and the geometric centre was marked with a board marker. A set of straws were filled with sand and capped with clay at the ends, which served as weights in the weighted cylinder. Each sand filled straw weighed approximately 25 grams. Using this apparatus, multiple trials were conducted releasing the PVC pipe with different number of straws each time (therefore, different masses). Slow motion video of this was captured using an iphone 6S at 240 fps. Please note from the experimental setup below that the is aligned along the inclined plane, where the point of release is plotted as the origin. Consequently, the entire motion is only along the.

2 Figure 1. Experimental Setup 3. HYPOTHESIS The experimental setup was used to observe the motion of the ball. The following hypothesis was then created. The initial observation was the cyclic nature of the motion, which included brief, repetitive periods of acceleration and deceleration. As it was rolling down an inclined plane, the overall motion was thought to resemble the motion of a hollow cylinder down an inclined plane with the repetitive accelerations and decelerations as aberrations. From this, the motion of the geometric centre of the weighted hollow cylinder was modelled as the sum of a linear motion graph adjusted by a sinusoidal equation. The following is the equation hypothesised by the above assumptions. Where is the angle of the inclined plane, is the displacement, is the variable time, is the radius of the cylinder, and is the length of the inclined plane. is an unknown constant. 4. EXPERIMENTAL DATA The following displacement-time graph was obtained from the experiment upon video analysis in Tracker Software. In this case, x refers to the displacement along x-axis of the geometric center, when the x-axis is aligned along the inclined plane (refer to Figure 1 for clarification), and where position of release corresponds to the origin.

3 Figure 2. Displacement-Time graph of weighted cylinder rolling down an inclined plane At first glance this appears to resemble the graph that would be generated using the hypothesized equation, but on closer inspection we find that the graph does not increase steeply at the end of every trough as it should. In order to test this, we plotted the derivative of this function, which was not like a sinusoidal curve at all, and rather resembles the curvature of a cycloid. In Figure 3, we can clearly see the velocity-time graph, which resembles a horizontally compressed, reflected cycloid aligned along an inclined plane. Here refers to the velocity of the geometric center of the cylinder along the inclined plane.

4 Figure 3. Velocity-time graph of weighted cylinder rolling down an inclined plane 5. MATHEMATICAL DERIVATION As the velocity during one complete rotation of the cylinder is maximum when the mass is at the bottom and decreases progressively until it comes to the top, after which it increases until it reaches the bottom again, it can be modelled using the equation of a cycloid. However, the cylinder is also accelerating due to the gravitational force acting on it as it moves down the inclined plane. This causes the motion to be modelled by the equation of a cycloid along a straight line inclined at an angle with the x axis. Moreover, as the average velocity of the geometric center of the cylinder (during each rotation) continually changes as the cylinder moves down the inclined plane, the time taken for each rotation decreases with each rotation. Thus, the x-axis distance between two consecutive troughs in a cycloid must decrease over time. This distance can be described by, where is the constant in the equation of a cycloid. In the experiment, this corresponds to the time taken by the cylinder to complete one rotation. The following calculations were performed to factor in the change in time of rotation into the equation of a cycloid. By Work-Energy Theorem, the loss in the cylinder s gravitational potential energy is equal to the overall gain in its rotational and translational kinetic energy. Thus,, where is the mass of the weight attached to the cylinder, and is the mass of the cylinder without the attached weight. As the moment of inertia of any object is equal to the sum of the moments of inertia of its constituent parts [1], the moment of inertia of a weighted cylinder can be found by summing the moments of inertia of a cylinder and a point mass [2]. Therefore,, where is the inner radius of the cylinder, and is its outer radius. As the angular velocity is the ratio of the translational velocity to the outer radius, the expression for the loss in gravitational potential energy can be written as: Solving for, This is the final velocity after integer multiples of rotations (determined by change in height of geometric center from initial position). As the change in is only influenced by the gravitational force acting on the cylinder, it remains constant down the inclined plane. Substituting Eq. in, and using the facts that and change in can be written as:,, the Simplifying yields: To calculate the time of the n th rotation of the cylinder, has been used., in this case, is the initial translational velocity of the cylinder before a particular rotation. Using,. As the distance travelled during one rotation of the cylinder is,

5 Use of the quadratic formula, and substitution of Eq. for gives as: Equating to corresponding (where is the constant in the equation of a cycloid), This equation gives as: As explained earlier, this governs the constant decrease in the x-axis distance between the two troughs of the cycloid. However, the y-axis distance, also governed by in the equation of a cycloid must remain constant. This necessitates a different constant that would govern the y-axis distance between a trough and crest in a cycloid. This distance can be interpreted as the change in velocity between the two different positions of the attached weight when it is at the top of the cylinder and when it is at the bottom. To calculate this change in velocity, the Work-Energy theorem was used once again. The change in height causing a change in the gravitational potential energy can be calculated using the location of the center of mass after half a cycle of rotation, added to the change in height due to the inclined plane. As the center of mass of the cylinder used can be expressed as lying at a distance of from the geometric center, the total change in height due to the change in the height of the center of mass is. Moreover, as the distance traversed by the cylinder during half a rotation is, change in height due to the inclined plane is (where is the angle of the inclined plane as defined earlier). Substituting this for in Eq., Simplifying gives: As this corresponds to in the equation of a cycloid (where is the constant in the equation of a cycloid which governs the vertical distance), To modify the equation of the cycloid so as to have different constants governing the x and y axis components of the graph, the cartesian form was derived from a slightly modified parametric form having different constants and for the x and y coordinates. Moreover, all the signs were changed so as to make the graph of the cycloid concave upwards rather than downwards. This modified cartesian equation is as follows: Substituting the values of (previously referred to as ) and into this equation of a cycloid and adding a linear term (using ), the final implicit equation modelling the studied velocity-time graph is obtained as: (note that for convenience, has been used instead of its corresponding expression) CONCLUSION The velocity-time graph can be modelled using Eq., where inputting the desired variables gives us the velocity at any given instant in time. We can infer from this that the velocity has a cyclical variation (evident from the term), which is non-uniform in nature owing to the effect of acceleration due to gravity. For further research, numerical methods can be used to calculate the velocity of the cylinder at any time. These values can then be matched with experimental data to obtain a visual proof

6 of Eq.. Moreover, the integral of the function can also be calculated in order to obtain an equation for the displacement vs. time graph. ACKNOWLEDGEMENT We would like to thank Mr. Gyaneshwaran, our physics teacher and mentor, for his valuable guidance and suggestions. REFERENCES [1] The Feynman Lectures on Physics Vol. I Ch. 19: Center of Mass; Moment of Inertia. The Feynman Lectures on Physics Vol. I Ch. 3: The Relation of Physics to Other Sciences, [2] OpenStax CNX, cnx.org/contents/unnozuzi@1.204:gw9kwukt@3/moment-of- Inertia-and-Rotation.