5-April-2017: Lab 9: Physics 4A Lab--Centripetal force with a motor

Eric Chong

Lab 9: Physics 4A Lab--Centripetal force with a motor

Lab Partners: Harvey Thai and Lynel Ornedo

Purpose: As the motor spins at a higher angular speed, the mass revolves around the central shaft at a larger radius and the angle θ increases. The goal of this lab is to come up with a relationship between θ and ω.

Introduction/Theory: There are two forces acting on the object: 1) gravity, and 2) the tension on the string acting as the centripetal force. The object must counteract the force of gravity and is also subject to the tension of the string, creating the angle θ observed. It is known that an object rotating at faster speeds would have a higher centripetal force. This is due to the object's inertia, its tendency to stay at its current state of motion. Objects with higher velocity require a stronger centripetal force in order to stay in circular motion. Since the object wants to fly outwards, the force required to keep the object inwards increases, thus the horizontal component of force increases. And this in turn increases the angle θ of the string. Our job is to find the relationship between θ and ω.

Procedure: At the start, professor already set up an apparatus for us that looks like this:

What we want to calculate is theta between the string and the vertical. To do this, we can calculate the sides of the right triangle present among the string, the vertical line, and the distance of the object from the vertical. Here is what it looks like:

Looking at this diagram, it is evident that we need to measure first the height of the apparatus, the length of the string, the length of the rod attached to the string, and the height the object as it is rotating. Measuring the length of the string, the length of the rod, and the total height is simple. For those measurements, we measured 158.5 cm, 74.9 cm, and 178.1 cm, respectively. These values will not change for each trial.

To measure the height of the rotating object, we used another stand with a rod sticking out with a piece of paper attached. We slowly raised the paper up until the paper started to touch slightly the rotating object. From there, we can measure the height of the rotating object for each trial run. After we got the height of the object, we can take the difference between the total height and the object's height in order to get the vertical leg of the triangle. Then, we can use that to solve for the angle with basic trigonometry, and also solve for the horizontal leg of the triangle. We do this because we want to use these values for our equation that we will develop in order to find the relationship between angular velocity and the angle. We ran a total of 6 trials, each with different angular velocities.

Before we get into the data measured for each trial, we wanted to write an equation for the relationship between the angular velocity and the angle. Here is how we did it:

This equation would help us achieve our "calculated" angular velocity, or "expected" angular velocity. We would also compare the values in our calculated angular velocity with our "actual" angular velocity, and we can get the actual values from measuring the time it takes for the object to rotate around the apparatus. We took the time for 10 revolutions, and we used that to calculate the period. After we calculated the period, we can divide 2pi over the period in order to give us the actual angular velocity.

Finally, here are our raw data typed in a spreadsheet format, with the calculations of the actual and expected angular velocity included, using the previous equation and the quotient of 2pi and the period:

(Note: there is a typo on the "h (cm)." It should be in meters.)

Looking at the data, the expected angular velocity is largely consistent with the actual angular velocity, as seen in the percent error. Therefore, our equation is consistent in calculating for the angular velocity, while also describing the relationship between the angle that the rotating object makes and its angular velocity.

Conclusion: The relationship between the angle and the angular velocity can be found through looking at the free body diagram of the rotating object. We can manipulate the components of the tension and gravity in order to solve for the angular velocity. It is also seen in the equation that the angular velocity also depends on the radius of the circle of the rotation, which is to be expected, given that an object's period also depends on the circumference of the circle. This equation is helpful, in that it gives us an "expected" value to look for when we compare it to the actual value of angular velocity.

Some sources of uncertainty could designate from the air resistance applied onto the rotating object. It is observed that the string is pulling back on the rod of the object, which indicates that there is some resistance to movement for the object. It could be due to the inertia, but since the object has a small mass, it is more likely for the resistance to come from the air resistance. Another source of uncertainty is the rod itself. Since the string is pulling back on the rod, it is likely for the rod to bend downwards, manipulating the actual angle that the object makes, and this uncertainty could also be accentuated at higher speeds, since the tension in the string increases. This is likely to make the measured values slightly lower, since the rod could possibly reduce the angles. Overall, these are the sources of uncertainty that is evident when we observed the rotating object.