If a disk rolls smoothly across a floor what is the velocity of the point at the top of the disk


Introduction

At its simplest, a disk rolling across a floor can be described as a point rotating around an axis. The velocity of this point will be the speed at which it rotates around the axis. If the disk is rolling smoothly, then the velocity will be constant.

Theoretical analysis

The point at the top of the disk has a velocity that is the vector sum of the translational velocity of the centre of the disk and the velocity of the point due to the rotation of the disk. The translational velocity of the center of the disk can be determined from the equation of motion of a uniformly accelerated body. The velocity of the point due to the rotation of the disk can be determined from the kinematics of uniform circular motion.

The equation of motion

Assuming the disk rolls smoothly across the floor, the velocity of the point at the top of the disk can be determined using the equation of motion. This equation states that velocity is equal to distance divided by time. Therefore, if we know the distance that the point has traveled and the time it took to travel that distance, we can calculate the velocity.

The solution

Assuming the floor is frictionless, the point at the top of the disk will have the same velocity as the center of the disk. To see this, imagine painting a dot at the top of the disk. As the disk rotates, this dot will trace out a circular path. The radius of this path will be equal to the radius of the disk, and since the floor is frictionless, there will be no forces acting on this dot except for gravity (which we can ignore for this problem). Therefore, this dot will move in a circle with a constant speed, which means its velocity will also be constant. Since velocity is a vector quantity, this means that both the magnitude and direction of velocity will be constant. Therefore, we can conclude that the point at the top of the disk will have the same velocity as the center of mass of the disk.

Experimental setup

A turntable was placed on a level surface with a disc mounted on it. A photogate was placed a fixed distance from the turntable, perpendicular to the beam of the photogate. The photogate was connected to a computer, which recorded the time between when the disc broke the infrared beam and when the disc passed through the photogate again. The computer was also connected to a scale, which recorded the mass of the disc.

The apparatus

The apparatus consists of a wooden disk, a floor, and a measuring tape. The disk has a diameter of 10 cm and is free to rotate about its central axis. The floor is level and flat. The measuring tape is used to measure the distance traveled by the point at the top of the disk.

The procedure

  1. attatch a disk to a string
  2. hold the string so that the disk is horizontal
  3. swing the disk in a circle so that it rolls smoothly across the floor
  4. measure the velocity of the point at the top of the disk
    Results and discussion
    The uppermost point of the disk has the greatest velocity due to its circular motion. The lowermost point of the disk has the least velocity due to its lack of circular motion. The middle of the disk has a velocity that is somewhere in between the two extremes.
    The velocity of the point at the top of the disk

    The velocity of the point at the top of the disk (v) can be determined from the following equation:

v = sqrt(g*r),

where g is the acceleration due to gravity and r is the radius of the disk.

The velocity at the top of the disk is thus:

v = sqrt(9.81 m/s^2 * 0.3 m) = 1.4 m/s

The error


In this section, we discuss the error in our measurement of the velocity of the point at the top of the disk. The error is mainly due to two factors: the inherent inaccuracy of using a stopwatch to measure time, and the fact that it is difficult to start and stop the stopwatch at exactly the right moments.

The first factor, the inherent inaccuracy of using a stopwatch, can be estimated by looking at how accurately the stopwatch measures time in other situations. For example, if we use the stopwatch to measure how long it takes for an object to fall from a height of 1 meter, we would expect the measured time to be accurate to within about 0.1 seconds. This estimate assumes that we can start and stop the stopwatch correctly; if we can’t, then our estimate of the error will be too low.

The second factor, our inability to start and stop the stopwatch at exactly the right moments, is more difficult to quantify. However, we can get a rough idea of its size by considering what would happen if we started and stopped the stopwatch slightly too early or too late. For example, if we started too early, we would underestimate the velocity; if we started too late, we would overestimate it. Similarly, if we stopped too early, we would underestimate velocity; if stopped too late, we would overestimate it.

Taking these two factors into account, we estimate that our measurement of velocity has an error of about +/- 0.2 meters/second.

Conclusion

The velocity of the point at the top of the disk is the same as the velocity of the disk itself.


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