Inclined Plane Handout FA23

INCLINED PLANE EXPERIMENT THEORY While on a frictionless inclined plane, an object experiences only two forces acting on it. One force is the normal force which or oriented perpendicular to the inclined plane itself. The second force is the force due to gravity which is oriented towards the Earth’s surface. The x- component of the force of gravity, oriented along the incline, causes the object to accelerate along the incline according to the equation ∑ i = 1 N F ix = ma x N x + W x = m a x F N cos ( 90 ° ) + F g cos ( 270 ° + θ ) = ma cos ( 180 ° ) F g [ cos ( 270 ° ) cos ( θ ) − sin ( 270 ° ) sin ( θ ) ] =− ma − F g sin ( θ ) =− ma F g sin ( θ ) = ma where F N is the normal force magnitude, F g is the force due to gravity magnitude, m is the mass of the object, θ is the angle of the inclined plane with respect to the Earth’s surface, and a is the object’s acceleration magnitude. The magnitude of the force due to gravity is defined to be F g = mg where g is the acceleration due to gravity magnitude. Using this definition, the object’s acceleration can be simplified as follows: mg sin ( θ ) = ma g sin ( θ ) = a where g is the magnitude of gravity’s acceleration, and θ is the angle of the inclined plane with respect to the table.

INCLINED PLANE EXPERIMENT Standard Deviation, Standard Error and Percent Uncertainty The STANDARD DEVIATION of a data set is calculated by subtracting the mean of the data set from each individual data point. This difference is squared to obtain a magnitude since we don’t care whether an individual data point is positive or negative. We then divide the sum of these squares by the number of data points ‘N’ (for small N we use N-1 in order to insure we do not overcount ‘N’.) We then take the square root of our result. σ = √ ∑ i ( x i − x ) 2 N ∨ N − 1 The STANDARD DEVIATION is also the square root of the VARIANCE. The STANDARD ERROR is the standard deviation divided by the square root of the number of data points. This gives the average variation for a single data point or the experimental error associated with a given variable. δx = σ √ N To find the percent uncertainty we divide our experimental error by the mean and multiply by 100%. ∆ x = δx x x 100% = σ √ n 1 x x 100%

INCLINED PLANE EXPERIMENT FIGURE 1 - The approximate set-up for today’s experiment. DATA ANALYSIS  Show the following calculations from the 5 ° incline trial: o Theoretical Acceleration  Magnitude  Magnitude Percent Error o Acceleration versus Time Plot  Experimental Acceleration Magnitude Percent Uncertainty o Position versus Time Plot  Experimental Acceleration Magnitude  Experimental Acceleration Magnitude Percent Uncertainty Angle Bracket Ring Stand

INCLINED PLANE EXPERIMENT DATA TABLE – ANGLE BRACKET RAMP ANGLE ZERO ERROR θ 0 ( ° ) 0 DATA TABLE – ACCELERATION SENSOR RAMP ANGLE θ±δθ ( ° ) ACCELERATION COUNT N () ZERO ERROR a 0 ( m / s 2 ) MEAN VALUE a±σ ( m / s 2 ) 5 ± 0.0087 0 0.753 ± 0.113 31 10 ± 0.0087 0 1.512 ± 0.044 29 15 ± 0.0087 0 2.125 ± 0.441 21 DATA TABLE – POSITION SENSOR RAMP ANGLE θ±δθ ( ° ) QUADRATIC COEFFICIENT A ±δA ( cm / s 2 ) 5 ± 0.0087 0.341 ± 0.010 10 ± 0.0087 0.724 ± 0.020 15 ± 0.0087 0.914 ± 0.035

INCLINED PLANE EXPERIMENT 5 ° Incline Trial – Position versus Time Plot Experimental Acceleration Magnitude a = 2 A a = ¿ 0.682 Experimental Acceleration Magnitude Percent Uncertainty ∆a = δA A x 100% ∆a = ¿ 2.933% Percent Difference (Experimental acceleration from position vs acceleration graphs) PD % = | a 1 − a 2 a 1 + a 2 2 x 100% | = ¿ 9.895% Percent Error (theoretical vs experimental) PE % = | T − E E | x 100% = ¿

INCLINED PLANE EXPERIMENT TABLES OF RESULTS RAMP ANGLE θ±δθ ( ° ) ACCELERATION MAGNITUDE a±∆a ( m / s 2 ) PERCENT DIFFERENCE PD ( % ) Position versus Time Plot Acceleration versus Time Plot 5 ± 0.0087 0.682 ± 0.020 0.753 ± 0.113 9.895% 10 ± 0.0087 1.448 ± 0.040 1.512 ± 0.044 4.324% 15 ± 0.0087 1.828 ± 0.070 2.215 ± 0.441 19.307% RAMP ANGLE θ±δθ ( ° ) THEORETICAL ACCELERATION MAGNITUDE a±∆a ( m / s 2 ) PERCENT ERROR PE ( % ) Position versus Time Plot Acceleration versus Time Plot 5 ± 0.0087 0.854 ± 0.0761 25.219% 13.413% 10 ± 0.0087 1.702 ± 0.153 17.541% 12.566% 15 ± 0.0087 2.536 ± 0.233 38.730% 14.492%

INCLINED PLANE EXPERIMENT No, because our theoretical acceleration magnitude was 0.854 m/s^2. In comparison, our data shows that the acceleration magnitudes on the position vs. time plot was 0.682 for 5 degrees, 1.448 for 10 degrees, and 1.828 for 15 degrees. These values are significantly different from our theoretical acceleration magnitude. 5. Identify the independent variable(s) for this experiment.  Be specific and use proper vocabulary . The independent variable is the angle of the ramp. 6. Identify the fixed variable(s) for this experiment.  Be specific and use proper vocabulary . The fixed variable is gravity 7. Identify the dependent variable(s) for this experiment. The dependent variable is the acceleration and position  Be specific and use proper vocabulary .  No CONCLUSION