Vector Addition of Forces, by Marcus Wade

Purpose:

The purpose of Vector Addition of forces was to study vector addition both by graphing, and by using vector components with trigonometry.

Equipment:

Tools that were used in this lab included a protractor, string, pulleys, varying mass (g), and a circular force table.

Procedure:

Circular Force Table

In this lab, three vectors were given, each with their own magnitude (g), and direction (angle in degrees). using a scale of 1cm= 20 (g) of mass, a vector diagram was constructed in which three separate force vectors were added.

fig 1

First, Vector A=200g at 0 degrees is graphed. Shown to the right. At an angle of 0 degrees, it simply follows the x axis for 10 cm (fig 1).

fig 2

Second,  Vector B=100g at 41 degrees is added to Vector A. Since its magnitude is 100g, the length of the vector is 5cm (fig 2).

fig3

Third, Vector C=150g at 132 degrees is added to the other Vectors, with a length of 7.5cm (fig3).

fig 4

Last, the resultant Vector R is connected from the tail of the Vector A to the head of Vector C. from graphing the resultant vector, its length appeared to be 12.5cm or 250g, at roughly an angle of 45 degrees (fig 4).













In order to obtain a more accurate depiction of Vector R, the x and y components were calculated using trigonometry for the three vectors that were added. The cos and sin of the vectors are used in order to get the magnitude of the x and y component vectors of A, B and C.

fig 5

Components and angles are shown in fig 5.

Where A= angle of vector

Sin(A)= ((y component)/(magnitude of vector))

(magnitude of vector)(sin(A))=(y component)

and also

Cos(A)= ((x component)/(magnitude of vector))

(magnitude of vector)(Cos(A))=(x component)

Plugging in all of the vector data to the equation, the x component of each vector are added to find R_x, and the y components are also added to find R_y.

R_x=200+100cos(41)+150cos(132)

R_x=175.1

R_y=0+100sin(41)+150sin(132)

R_y=177.1

Using our new values of R_x=175.1 and R_y=177.1, the exact magnitude and angle of Vector R can now be calculated.

fig 6

(Magnitude of Vector R)=(175.1²+177.1²)^1/2

(Magnitude of Vector R)=249g

(Angle of Vector R)=tan^-1(177.1/175.1)

(Angle of Vector R)=45.3 degrees

Measurements of resultant vector are shown in fig 6.

fig 7

The negative of Vector R then needed to be taken in order to create equilibrium between the forces. To do this, the vector components are simply made negative, and then take the new angle (225 degrees). Lets call this Vector -R (fig 7).

Setup:

Once the components of all the vectors had been obtained, the vectors were to be physically plotted on the circular force table. On the first holder, start with a force of 200 g at 0 degrees, then add a force of 100 g at an angle of 41 degrees on the second holder. Continuing,the third vector was added with mass 150 g at 132 degrees on the third holder.

What happens when you place a mass on the fourth holder equal to the magnitude of the resultant vector? When Vector -R (which is equal in magnitude to the resultant vector, but opposite in direction) is plotted physically and placed on the fourth holder, it creates equilibrium between the forces on the circular force table.

 Pictured in fig 8 is the equilibrium resulting from addition of all four force vectors.

The vector addition was then checked on a website (http://phet.colorado.edu/en/simulation/vector-addition). We input all of our vector information to find the resultant vector, and it seemed to match very well with our data (fig 9).

fig 9

Conclusion:

In this lab, one was able to learn about the addition of vectors both graphically and through using components. Graphing the vectors seemed to be a reasonably accurate way to estimate the magnitude and direction, but using the vector components was able to give one a clear and precise answer to our resultant vector. Possible sources of error in this lab would include having to estimate a weight in grams to balance, or even not setting the angles precisely on the circular force table.

Acceleration of Gravity on an Inclined Plane, by Marcus Wade

Introduction:

In this lab, position vs. time data points will be taken of a cart moving up and down an incline. The motions of the cart traveling up, then down the incline will be treated as separate, then compared. Because the force of friction acts with the motion of the cart on the way up ramp, and acts against the motion on the way down the ramp, the average of the two accelerations will be taken with the following ratio:

gsin(A)= (a₁+a₂)/2

where g= gravity, A= angle of incline, a₁= acceleration up incline, and a₂= acceleration down incline

Purpose:

The purpose of Acceleration of Gravity on an Inclined Plane was to determine the effect of gravity on an object traveling up and then down an inclined plane, while gaining further experience with the graphical analysis software.

Setup:

Tools used included the logger pro software, a motion detector, aluminum track, ballistic cart, carpenter level, meter stick, and a wooden block. The aluminum track was setup at an incline with the wooden block, and then carefully leveled. We then measured the change in height of the two sides, to the horizontal length of the inclined ramp (pictured in fig 1). The angle of the ramp was then calculated with the following ratio:

fig 1

A=angle of incline on ramp

Tan^-1((y₂-y₁)/(x₂-x₁))=A

Tan^-1((12.9cm-6.7cm)/(228.2cm))=A

A=1.56^o

Procedure:

 After the setup was completed, the trial runs were performed where the cart was tossed up the ramp and then fell back down. It took a few practice runs in order to obtain a trajectory that did not come within 50 cm of the motion detector. Three trial runs were completed first for the inclination angle of 1.56 degrees, and then the process was repeated for an inclination angle of 3.6 degrees.

What type of curve do you expect to see for x vs. t and v vs. t? I expect that the curve of x vs. t would be a parabola with the left side having a more rapid change in slope than the right because of friction acting with motion of the cart on the way up and against it on the way down. I also expect the graph of v vs. t to be linear with different slopes on the motion up and down the ramp (because of the effect of friction that changes the acceleration).

Shown below are the position vs. time and velocity vs. time graphs of our first trial at an angle of 1.56 degrees.

fig 3 position vs time graph

fig 4 velocity vs time

Position vs. time graph

Velocity vs time graph, note how the motion on the way up the ramp has a greater change in velocity than the motion down the ramp due to friction

In order to obtain our values for a₁ and a₂, a linear fit was applied to the negative velocity portion of the v vs. t graph for a₁, as well as a linear fit to the positive velocity of the v vs. t graph for a₂. The slope from our linear equation then became our separate accelerations




Data:

Our data for each of the trial runs is shown in the spreadsheet. The first table is for a ramp angle of 1.56 degrees, and the second for an angle of 3.6 degrees. a₁ is the cart's acceleration up the inclined ramp, and a₂ is the cart's acceleration down the track.

Conclusion:

Once the data had been collected, verification of our numbers compared to the accepted value of acceleration due to gravity was needed. Since the force of friction acts with the force of gravity on the way up the ramp and against it on the way down, the average of the two accelerations will be taken to equate our experimental value of acceleration due to gravity. The formula used to calculate G_exp was:

G_expsin(1.56)=(a₁+a₂)/2

-Trial one of 1.56 degree incline

G_expsin(1.56)=(0.33 m/s²+0.18 m/s²)/2

G_exp=9.4m/s²

In this lab the group was able to determine with reasonably close accuracy the effect gravity has on objects that are on inclined planes. The percentage differences they calculated in this lab seemed to match well with the accepted 9.8 m/s². The most inconsistent value was 9.4 m/s² at a 4.1% difference. Potential sources of error in this lab would have included various factors such as the friction between the car and the track, and performing the linear fit to the velocity vs. time graphs in different domains.