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Introductory Mechanics Experimental Laboratory
Gravity Slide
Goals: Observe motion in one dimension. Use an electronic measuring device to acquire data. Convert numeric data to a graph.
APPARATUS Galileo Galiliei (1564-1642) used an inclined plane with minimal friction to observe motion due to gravity. The track in this experiment is more technically sophisticated, but the idea is the same as it was 400 years ago. The height of the incline can be adjusted to change the angle of the slope. A cart moves up and down the incline, and an acoustic motion detector senses the position of the cart at a sequence of times.
The data will be recorded with Logger Pro software. When the software is open there is a window on the left side of the screen with columns labeled time, position, and veloc- ity. On the right side there are two graphs, one to show position versus time, and the other to show velocity versus time.
THEORY An object sliding down an inclined plane without friction is subject to a constant accel- eration ( a ). The acceleration is due to the gravitational acceleration ( g ), but the acceler- ation on the slide is reduced because the track is at an angle. The full gravitational acceleration would only occur if the object fell straight down and the object would then be in free fall. The gravitational acceleration near the earth’s surface is g = 9.8 m/s^2.
The reduced acceleration due to angle of the track can be expressed as
(EQ 1)
where θ is the angle of the track compared to a horizontal surface. The sine of the angle can be found from two heights, h 1 at the higher point and h 2 at the lower point, and the distance L between them along the slope.
(EQ 2)
a = g sinθ
sin θ
h 1 – h 2 L = -----------------
2 Gravity Slide
So, for our track the actual acceleration for a cart should be
(EQ 3)
The acceleration ( a ) of an object is defined as the rate of change of the object’s velocity ( v ). Velocity is the rate of change of position ( s ) of an object. The rate of change of a quantity is equal to the amount of change of the quantity divided by the amount of time that change required.
Suppose an object were at position s 0 at time t 0 , position s 1 at time t 1 , and at position s 2 at time t 2. The average velocity during the time from t 0 until t 1 and from t 1 until t 2 would be
(EQ 4)
The average acceleration can be determined from the average accelerations
(EQ 5)
These are only average velocities and accelerations, since the velocity will be changing during that time. The smaller the time interval considered the closer your calculated velocity is to the exact velocity at which you were moving during that interval. Instanta- neous velocity is the velocity which would be calculated if an extremely small (nearly zero) time interval were used.
DATA COLLECTION 1. Set the height at the high end of the track so that it is appoximately 20 cm above the table. Measure the length ( L ) of the air track between two points on the track and the heights ( h 1 , h 2 ) of the track at the two points used to measure the length.
2. Confirm that acoustic motion detector is set to the “cart” setting. Open the Logger Pro software on the computer. Right click on each graph and select graph options , select the tab axes options , and confirm that the scaling is set to auto scale. 3. Confirm that the flag is magnetically attached to the cart. Practice gently sliding the cart up the track from the bottom so that it goes about halfway up the track. If it is pushed too hard it will derail, and it needs to be caught at the bottom. When the cart falls it can cause misalignment which increases friction. 4. Start data collection by clicking on the collect button on the software. Gently push the cart up the ramp when you hear a clicking sound and catch it at the bottom. 5. Review the graphs in Logger Pro. The position versus time graph should be a smooth parabola, and the velocity versus time graph should be a straight line. Repeat step 4 until both curves are reasonable, and check with the TA to verify that they are. 6. Copy the data from Logger Pro into a table or as a .csv file on a USB drive along with the dimensions in step 1. 7. Repeat steps 1 through 6 but with the high end about 15 cm above the table. 8. Repeat steps 1 through 6 but with the high end about 10 cm above the table.
a g
h 1 – h 2 L = ⎝⎛ -----------------⎠⎞
v 1
s 1 – s 0 t 1 – t 0 = --------------- and v 2
s 2 – s 1 t 2 – t 1 =---------------
a
v 2 – v 1 t 2 – t 1 =----------------
