3) Before we learn the aspects of angular acceleration, we need to know how it is measured. Angular motion is measured when a point is rotating around a fixed point or a fixed axis. In linear motion, the motion of an object is measured in x and y-coordinates. This is inconvenient for a rotating object. For rotation, we measure in angular coordinates where a point P is a distance away from the center O. As a fixed point rotates, the angular displacement represented by delta theta, is the angle through which it moves. Theta is measured in radians. Using radians lets us relate the linear distance around the circle called the arc length equal to the angular distance theta and the radius R. In linear motion we learned about linear displacement, linear velocity, and linear acceleration. These three quantities carry the same aspect for linear motion, which are angular displacement, angular velocity, and angular acceleration. Linear velocity can still be measured in rotational motion which we call tangential velocity. The difference is tangential velocity is the linear velocity of one point on the rotation object while angular velocity is the rate of rotation of the entire object. The relationship between linear velocity and angular velocity is linear velocity is equal to the angular velocity times the radius. This is the same relationship for angular acceleration. The change of angular velocity over time is the angular acceleration. The relationship between linear acceleration and angular acceleration is linear acceleration is equal to the angular acceleration times the radius. Torque is a force applied to a rotating object that is perpendicular to the motion. The direction of torque is in the same direction of angular acceleration. So the torque is equal the force tangent times the radius. Since the force is equal to mass and angular acceleration the equation is converted to the moment of inertia and angular velocity squared. The moment of inertia is the sum of the mass and distance from the origin of the rigid body going in rotational motion. Since there is more than one point of the mass going in rotation, we have to sum all the point masses.
4) We will be using the Pasco rotational sensor in order to record the angular acceleration through Logger Pro. This device simply compresses air between the two disks to cancel out friction as the dick rotates. We will attach a string to the center of the disk and clamp a pulley. As a hanging mass is attached to the other end of the pulley, the disk will simply oscillate clockwise and counter clockwise as the pulley attached to the disk applies torque. We use the hanging mass as torque to simply calculate the angular acceleration since force and acceleration are related according to Newton's 2nd Law. With the string wrapped around the torque pulley and the hanging mass at its highest point, start the measurements and release the mass.
5) Here is the data we recorded using the Logger Pro software in lab part one:
.In lab part two, we use the data table to input the values to find the moment of inertia of the disks (or disk combination) in each experiment.
6) Here are the angular displacement and angular velocity vs time graph. They are all similar to each other but with different acceleration rate for each experiment.
Here are the six experiment graphs of what our assumptions we wanted.
7) In lab part 1, the direction is the first detail we must point out before graphing. As the angular acceleration is negative while the angular velocity is negative and increasing, that means the disk is rotating clockwise. While the angular velocity is decreasing speed in the clockwise direction, the angular acceleration is in the positive direction.
This intuition applies to all the six experiments, no matter if we change the masses or the size of the torque pulleys. In experiment 1,2, and 3, the changing effects is the hanging mass. The hanging mass is used in the experiment as a tangent force. The only force applying to the hanging mass is the force of gravity. As the hanging mass oscillates vertically, torque is applied to the disk and causes the disk to rotate. In the first three experiments, we increased the hanging mass. The more mass applied, more force is applied to the disk, meaning higher acceleration is applied.
Now we will look at experiment 1 and 4 of the effect of changing the radius and which the hanging mass exerts a torque. The hanging mass in experiment 1 and 4 is the same on the circular top disk. The only difference is the radius and the mass of the torque pulley. Our assumption of the angular acceleration of experiment 4 is greater than experiment 1.
Lastly, we will look at the effect of changing the rotating mass. The torque pulley and the disk will be counted together as one mass to calculated the differences in experiment 4, 5, and 6. The difference between the experiments is the mass of the rotating disk. We will determine the angular acceleration in which it rotates faster if the mass is greater or lesser. According to our data, the lightest mass in the experiment, has a faster angular acceleration than the heavier ones with the same torque they all have.
In lab two, before we conclude the moment of inertia of each experiment, there is a frictional torque by the mass of the disk rotating. The angular acceleration when the hanging mass goes down does not equal to the angular acceleration when the hanging mass goes up according to our data. If they did both equal each other, then the frictional torque would equal zero. The instructions instruct us to derive the frictional torque equal to zero by the following steps.
The angular acceleration multiplied with the moment of inertia is equal to the difference of the torque of the string and frictional torque. Same method for the angular acceleration when it goes up but the torque of the string and frictional torque are added together. Then we use Newton's 2nd Law of the hanging mass and substitute those equations into the first two. Then we derive the frictional torque where the moment of inertia of the disk comes from.
Frictional torque equals (mr^2 + I_disk)(1/2)(|Angular acceleration_up| - |Angular acceleration_down|)
If the angular acceleration up and down equal to each other, they cancel each other to zero and multiply the rest to zero.
8)
According to our data on the first three experiments, as we increased the hanging mass, the angular acceleration increased. It makes sense since the tangential force and the angular acceleration vector are parallel to each other and relate. So as the force increases, the angular acceleration increases.
In the first experiment, the radius of the torque pulley read 1.131 rad/s/s and experiment four read 2.1725 rad/s/s. So as the radius increases with the same tangential force, the angular acceleration. The mass of the rotating disk are the same in experiment one and four.
In experiment four through six, we changed the rotating mass disk and each angular acceleration give us a different result. The heavier the rotating disk is applied from the torque, the angular acceleration is slower than the lightest rotating disk. This makes sense since the heavier the object to push to the same acceleration of the lighter object, more force is required.This is our conclusion for our lab in part 1 to demonstrate the different factors of angular acceleration.
In lab part two, we conclude the moment of inertia increase when applied torque increases.
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