PROCEDURE:
For this lab we used a contraption that shoots a small ball into a small block hanging from a string in a pendulum fashion. When the ball shoots into the block, it moves the block up to a certain degree, which is measured by a hanging wire that is moved by the block. The wire stays at the highest angle to measure it afterwards.
Conservation of momentum can be applied to the portion of the experiment where the ball is shot into the block, to when the block and ball move together as one mass with one velocity. The equation for this is mball*vo= (mball+mblock)*vf. Once the ball is shot into the block, conservation of energy can be applied to the system of block and ball as a whole. The conservation of energy cannot be applied to the portion before the collision because some energy is lost during the collision. The equation for conservation of energy immediately after the collision is
1/2(mball+mblock)vo^2 = 1/2(mball+mblock)vf^2 + mgh.
Since we will be measuring the angle and height at which the ball and block move the highest, we know that the final velocity will be zero at this point. Therefore:
1/2(mball+mblock)vo^2 = mgh.
Additionally, vo^2 is equal to vf from our conservation of momentum equation. Height in this equation would be the length of the string minus the vertical distance that the block is at at its highest point, or L (length of string) - L cosθ => h = L(1-cosθ). (θ is the angle at which the block moves). We can use our second equation to solve for the initial velocity of the ball and block system, and plug that back into our conservation of momentum equation as final velocity, to find the initial velocity of the ball as it is first shot out:
Before the lab, we measured the mass of the ball to be 7.63g +/- 0.01g; mass of the block to be 80.9g +/- 0.1g. From the experiment we found θ to be 16.3 degrees, or 0.2854 rad, and the length of the string (L) to be 20.5 cm. Plugging in this to our equation (and converting to kg and meters) for conservation of energy, we find initial velocity of the system to be 0.402 m/s. Using this as final velocity for conservation of momentum, we calculate the initial speed of the ball to be 0.439 m/s.
PROPAGATED UNCERTAINTY:
To find propagated uncertainty in the velocity of the ball, we found the partial derivative of the velocity with respect to each of the following: mass of the ball, mass of the block, length of the string, and angle that the pendulum moved. Each partial derivative shows how much that measurement throws velocity off by, which we later multiply by a value of how uncertain we were with our measurements. For the mass of the ball, we chose an uncertainty of 0.01 g; for the mass of the block we chose an uncertainty of 0.1 g, for the angle we chose an uncertainty of 0.1 degrees, or 0.0017 radians; for the length of string we chose an uncertainty of 0.1 cm.
The following images are the calculations for all the propagated uncertainties:
 |
| Equations for partial derivative of velocity of ball, with respect to mass of block, mass of ball, length of string, and angle, respectively. |
 |
| Plugging in the numbers for partial derivative with respect to mass of block, then partial derivative with respect to mass of ball. |
 |
| Plugging in the numbers for partial derivative with respect to length of string, then partial derivative with respect to angle. |
.JPG) |
| We multiplied the partial derivative for each measurement by the value of how uncertain we were with our measurements. Plugging in the numbers and adding together each error, we get a propagated uncertainty of 5.08 cm/s, or 0.0508 m/s. |
CONCLUSION:
If we convert our value for the velocity of ball from 0.439 m/s to 43.9 cm/s, we can see that our value may fluctuate to 5.08 cm/s more or less than this velocity.
No comments:
Post a Comment