27-April -2015: Ballistic Pendulum Lab

The purpose of this lab was to find the initial velocity of some mass in a ballistic pendulum by combining both conservation of momentum and energy.

Setup:

Fig. 1
The setup

Unlike most of our other experiments, this lab was initially setup and ready to go the day of our lab. The apparatus consisted of a ballistic pendulum to which a spring gun was aimed at. From the spring gun a small mass projectile would be shot out. When done correctly, the mass would become lodged in the pendulum, causing it to swing upwards and to the right. The angle displacement of the pendulum could then be read by a protractor attached to the apparatus. Beginning at zero degrees from which the pendulum was stationary, the protractor would swing upwards as the pendulum pushed, and would then stay in place as the pendulum swung back. This would give us the angle of displacement. (see Fig. 1 and Fig. 2 for setup)








Measuring Data:
Once the projectile was loaded into the spring gun, we cocked the spring back, allowed the ballistic pendulum to be completely still, and FIRED! As expected, the pendulum swung to the right and was displaced by some angle. All we could measure from this was the angle of displacement, which measured to be 17.2 degrees (+/- 0.1 degree) and the length of the string(L) which was 21.7 cm (+/- 0.1 cm)

Fig. 2
The ball was lined up so that it would fit snug into the pendulum.

Calculating Initial Velocity:
We only had two equations at our disposal that we could have used to calculate the initial momentum of the projectile: Conservation of Momentum and Conservation of Energy equations. (NOTE: We were given the mass of the projectile(7.63g +/- 0.01g) and the mass of the ballistic pendulum(80.9g +/- 0.1g) For the conservation of momentum, we were missing the initial velocity of the projectile and the final velocity of the projectile and pendulum (see Fig. 3)

Fig. 3
We had to come up with an expression for the final velocity of the system.

Since our end goal is to find the initial velocity of the ball, we had to come up with an expression for the final velocity of the ball and pendulum. This is where our conservation of energy equation comes into play. Conceptually, we know that the initial energy will only consist of the kinetic energy of the ball and pendulum, and the final energy (gravitational potential energy) we would consider at the maximum height of the pendulum. (see Fig. 4 and Fig. 5)

Fig. 4
To calculate the height at which the pendulum went, we had to do some trigonometric work.

Fig. 5
Our final expression for the velocity of the system after initial impact.

Once we had our expression for the velocity of the ball and pendulum, we could plug that back into our original momentum equation. (see Fig. 6) At this point we could simply plug in our values for the mass of the ball, mass of the pendulum, length of the string, and the angle at which it was swung. This would eventually give us an initial velocity of 5.06 m/s. (see Fig. 7)

Fig. 6
Our final expression for the initial velocity of the ball.

Fig. 7
We found our initial velocity to be 5.06 m/s BEFORE finding our uncertainty.

Calculating the Propagated Uncertainty:

Although we had our value for the initial velocity of the projectile, we had to account for the fact that all of our measurements had some uncertainty. To do this, we did propagated uncertainty (see Fig. 8 for all calculations).

Fig. 8
Calculating the uncertainty in our entire expression.

Our final answer came out to be 5.060 m/s +/- 0.052 m/s for the initial velocity of the ball. The only major sources of uncertainty occurred when we had to read the measurement of something with our naked eye (ruler for length and protractor for angle). Other than that, however, our answer came out to be fairly accurate.