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Interactive Free-Body Diagram (FBD): Statics

A point mass m is on a ramp inclined by angle we wish to maintain static equilibrium for this point mass on the ramp by applying a force F inclined by angle , as shown in the diagram below. The static coefficient of friction between the point mass and ramp is S.

g Y X

F

m

With the point mass assumption, there can be no box rotation by definition. The statics freebody diagram (FBD) for the point mass box is shown in the figure below:

F

W Ff

N

In the XY coordinate system shown (X is aligned with the ramp), these four force vectors are expressed as: W

W

sin cos

F

F cos F sin

N

0 N

Ff

Ff 0

The primary purpose of this exercise is to allow the user to change various system parameters, calculate the required force F for static equilibrium, see the effects on the FBD, and feel each of the F

0 . It is forces. From the free-body diagram, we apply the vector equation of static equilibrium, convenient to use the XY coordinate system shown in the first diagram above: X is along the ramp F

0 yields: direction and Y is normal to the ramp.

X:

F Y :F

x y

F cos F f

W sin

0

N F sin W cos

0

S N , upwards along the ramp to resist motion down the ramp, as shown in the free-body diagram; N is the normal force of the ramp acting on the point mass. The weight force is W

mg , trying to move the point mass down the ramp. Substituting these given parameters into the statics equations yields:

The static friction force is F f F cos S N F sin N

mg sin mg cos

These are two linear equations coupled in the two unknowns F, N. It is convenient to solve this using a matrix approach (high-school algebra is fine too). The matrix-vector expression of the above equations is:

cos S F mg sin

sin 1 N mg cos

The matrix-vector solution for the unknowns is: F cos S

N sin 1

1

mg sin 1 1 S mg sin

cos mg cos sin mg cos

Where

cos S sin is the matrix determinant. The solution is:

F

mg sin S cos cos S sin

N

mg cos cos sin sin cos S sin

Using the cosine sum-of-angles formula from trigonometry ( cos a cos b sin a sin b

cosa b ), the solution for N simplifies: N

mg cos cos S sin

To satisfy static equilibrium, the above expression for F must be applied; otherwise dynamics motion will result. An interesting side problem is to apply zero force F and calculate the maximum ramp angle for static equilibrium for a given m and S. In this case the statics equations are:

S N N

mg sin mg cos

The Y equation yields the unknown normal force: N yields the maximum angle for static equilibrium:

mg cos ; substituting this into the X equation

MAX

tan 1 S

Note this result is independent of mass m. For this result, F = 0 will yield static equilibrium for any MAX . 0 m 10 0 1 0 0 Computer sets: Visualize: equilibrium.

User sets:

m, S , , and

90 90

g = 9.81 m/s2, (down, not in the ­Y direction unless =0). Free-body diagram with forces to scale. In the alternate problem, MAX for static

Numerical Display: F, N, Ff, and W; plus MAX for alternate problem. User Feels: Forces F, N, Ff, or W (user chooses). Joystick should display the vector forces to the user's hand, either in horizontal/vertical or XY coordinates. Example: When the user enters m = 80 kg,

35 ,

50 , and

0.15 , the force for static equilibrium is F = 381.52 N and the associated normal force, friction force, and weight are N = 544.12 N, Ff = 81.62 N, and W = 784.80 N. For this coefficient of static friction, the maximum angle for static equilibrium with F = 0 is MAX

8.53 .

Comprehension Assignment: Once you get the `feel' for this simulation, run the program several times to collect and plot data: for m = 10 kg and a fixed value of the static coefficient of friction S, vary the ramp angle and force angle over their allowable ranges and determine the resulting forces F, N, Ff, and W. Plot F, N, Ff, and W vs. . For each case, there will be different curves for different values. Repeat these plots for various values of S over its allowable range. For the alternate problem with F = 0, plot MAX vs. S. Discuss the trends you see in all cases ­ do the results make sense physically?

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