Lab 3: Acceleration of Gravity on an Inclined Plane

Purpose: 1. To find the acceleration of gravity by studying the motion of a cart on an incline.
                2. To gain further experience using the computer for data collection and analysis.

Equipment Needed: Windows based computer with Logger Pro software, motion detector, ballistic cart,  aluminum track, wood blocks, meterstick, small carpenter level.

Introduction: 
      In this laboratory you will use the computer to collect position (x) vs time (t) data for a cart accelerating on an inclined track . By comparing the acceleration of the cart when moving up and down the track, the effect of friction can be eliminated and the acceleration due to the effect of gravity alone can be found.
      Since the force of friction acts with the force of gravity when the cart is going up the track and against the force of gravity when the cart is going down the track, we can average the slightly increased acceleration (when going up) with the slightly decreased acceleration (down) to obtain an acceleration that depends only on the force of gravity. If we call g the acceleration due to gravity when an object is in free fall, then the component of this acceleration along the track is gsinθ where θ is the angle of inline for the track. Thus:

                                                               gsinθ=(a1 + a2) / 2

Where a1 and a2 are the acceleration of the cart up and down the inline. In this lab we will measure acceleration by looking at the slope of the v vs t curve for the cart.


Analysis:

When cart goes up, the force of friction is negative (we assume that the direction of velocity is positive). If the acceleration of friction is f, then:
                                                                 a1 =  | gsinθ | + | f |

When cart goes down, the force of friction is positive (we assume that the direction of velocity is negative). If
the acceleration of friction is f, then:
                                                                  a2  =  | gsinθ | - | f |

So: (a1 + a2) / 2 =（ | gsinθ | + | f |  + | gsinθ | - | f | ）/ 2 = (2 | gsinθ |) / 2 = | gsinθ |



Determine the θ:

a & b: The distance between aluminum track's endpoints to the desktop.
c: The length of the aluminum track.

In order to get the inclination angle θ, we measured the length of a,b,c:
a=6.3cm
b=13.2cm
c=227cm

so:
d=a-b=13.2cm-6.3cm=6.9cm
sinθ=d/c=6.9cm/227cm≈0.03
θ=arcsin(0.03)≈1.74°


Part 1:

R-T and V-T graphs 

According to our position vs. time graph (position is the distance from detector to cart) :
Position is decreasing between 0.6-3.2s, so the cart is going up;
Position is increasing between 3.2-5.8s, so the cart is going up.

On velocity vs. time graph:
 Velocity is negative between 0.8-3.2s, the cart is going up:
 Velocity is positive between 3.2-5.8s, the cart is going down:

Because the acceleration is the slope of velocity vs. time graph, we use the linear function to fit the curve and find the slopes when cart goes up and cart goes down.

When cart goes up:
a1 = 0.3 m/s^2

When cart goes down:
a2 = 0.25m/s^2

So:
                                                          gsinθ = (a1 + a2) / 2
                                                     g × 0.03 = (0.3m/s^2+0.25m/s^2)/2
                                                     g × 0.03 = 0.275m/s^2
                                                                g = 9.2m/s^2


Then we repeat our experiment two more times to get the average data:

Our gravity data when  sinθ=0.03

Part 2:

Then we use a lager block of wood to increase the angle of inclination of the track:

a=5.1cm
b=19.5cm
c=227cm

so:
d=a-b=19.5cm-5.1cm=14.4cm
sinθ=d/c=14.4cm/227cm≈0.06
θ=arcsin(0.06)≈3.64°

Our gravity data when  sinθ=0.06

Conclusion:
  According to two experiments, we found when θ is larger, our experimental data are closer to actual data(0.5% diff compare to 8.2% diff). The reason is that when θ is larger, the motion of cart is closer to free fall, which is influenced less by the disturbance. 
The causes of error: 
(1) Air resistance also against the motion. 
(2) Our table is not horizontal, so our θ is not precise enough. 
(3) The error of the equipment and the error when we read the data.
In this lab, we learned acceleration along the track is gsinθ where θ is the angle of inline for the track. we can use this property to estimate the gravity. We also learned how to control the variable to get another group of data, then try to think abut what cause the difference.