The purpose of this lab is to verify the impulse moment theorem. This theorem proved that an impulse is equal to the change of momentum which is equal to the the integral of a force equation as a function of time. To verify this theorem, there will be three total collisions in this lab: 2 elastic and 1 inelastic.
Setup:
Since there are three total collisions there will be three different setups. The first setup will serve to observe collision forces that change with time. We clamped a dynamics cart to a rod clamped to a lab table and extended the spring plunger on the dynamics cart. We then mounted a force sensor on another dynamics cart and placed a rubber stopper over the hook of the force sensor. When the two carts would collide the stopper would collide with the plunger of the stationary cart. Lastly we placed a motion detector at the other end of the track. The set up can be seen below:
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The setup for the next collision was the same as the first except this time we added 500 g to the cart:
For the last collision, the setup differed. We left the 500 g mass on the cart but we attached a nail to the stopper on the force sensor. We then replaced the dynamics cart with a wooden block and attached a ball of clay to it at the height of the nail. Now, when the cart hits the wall, it would come to rest at the block. Pictures of the setup can be seen below:
You will notice the 500 g mass is not in the picture but it was used in the experiment. Also we thought it would be fun to make our clay into the shape of a little man.
Procedure:
For the first collision:
1) We completed the setup as mentioned above
2) We made sure the software used to record our data was set up to record a push on the force probe as a positive force and velocity toward the motion detector as positive.
3) We calibrated the force probe using the hook screwed into the force sensor
4) Lastly it was important to be able to push the cart toward the plunger while making sure without our hands getting between the motion sensor and the cart
5) Finally, we started collecting data, pushed the cart, released it, and let it collide. We did this until we got a good set of data.
The procedure for the second collision was exactly the same as the first except this collision took place with an addition 500 g on the pushed cart.
For the final collision:
1) We completed the setup as mentioned above
2) We again made sure the software used to record our data was set up to record a push on the force probe as a positive force and velocity toward the motion detector as positive
3) The force prove was still calibrated from the first two collisions
4) Lastly, we started collecting data, pushed the cart, released it and let it stick to our clay on the block.
Results:
For the first collision:
The force is not constant. However the impulse-momentum theorem states the amount of momentum change for the moving cart is equal to the amount of net impulse on the cart. This collision was meant to test that idea. We measured the impulse acting on the cart by constructing a Force vs Time graph and taking the area under it. We measure the change in momentum by knowing its mass and measuring its velocity before and after the collision using the motion detector.
The mass of the cart: .432 kg
This was our Force vs Time graph:
As you can see from the graph the impulse was found to -0.4014 s*N
We can calculate change in momentum by looking at the velocity before and after the collision. At 1.66 s the velocity= .494 m/s. At 1.86 s, after the collision took place, the velocity = -.454 m/s. So:
change in momentum= mvf - mvi = (0.432*(-0.454)) - (0.432*0.494) = -0.409 N*s
For the second collision:
The graph below is the Force vs Time graph we were able to construct with our gathered data. We again measured the impulse acting on the cart by taking the area under the curve:
The mass of the cart + 500 g = (.434+.500) = 0.934 g
As seen in the graph the impulse was found to be -0.9374 s*N
We again can calculate change in momentum by looking at velocity before and after the collision. At 1.50 s the velocity = .515 m/s. At 1.78 s, after the collision took place, the velocity = -.462 m/s. So:
change in momentum= mvf - mvi = (0.934*(-0.462)-(0.934*0.515)= -0.913 N*s
For the final collision:
This collision was inelastic. When the moving cart hit the clay, it stuck to it instead of bouncing back like the other collision.
The graph below is the Force vs Time graph we were able to construct with our gathered data. We again measured the impulse acting on the cart by taking the area under the curve:
The mass of the cart + 500 g = (.434+.500) = 0.934 g
As seen in the graph the impulse was found to be -0.3134 s*N
We again can calculate change in momentum by looking at velocity before and after the collision. At 1.18 s the velocity = .490 m/s. At 1.38 s, after the collision took place, the velocity = 0 m/s. So:
change in momentum= mvf - mvi = (0.934*(0)-(0.934*0.327)= -0.305 N*s
Conclusion:
As you can see from the above table the collisions proved a success as the percent error was very small and the values were around the same numbers. However in the last collision, the percent error was the highest. But it was still in the acceptable range. So in, conclusion, our collisions proved the impulse momentum theorem.
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