Question

learning goal: to understand how to construct a free - body diagram for an object that can be treated as a particle. the free - body diagram is a fundamental tool used in engineering mechanics. it is simply a sketch that shows the particle \free\ from its surroundings with all forces that act on the particle. by correctly constructing a free - body diagram, one can account for all of the forces in the equations of equilibrium. it is often helpful to draw an enclosing circle (or loop) around the object(s) of interest and account for (a) forces in cables cut by the circle, (b) reaction forces between the object and any supports at the boundary of the circle, and (c) the weight of the object(s) enclosed by the circle. figure select the elements from the list and add them to the canvas setting the appropriate attributes. press ctrl+m to get to the main menu. no elements selected

Answer:

To construct the free - body diagram for particle \( C \):

Step 1: Identify the particle of interest

The particle we are interested in is \( C \). So we first draw a circle (or a dot) to represent particle \( C \).

Step 2: Identify the forces acting on particle \( C \)

• Force from cable \( AC \): Cable \( AC \) exerts a tensile force on \( C \). The direction of this force is along the cable \( AC \), pulling \( C \) towards \( A \).
• Force from cable \( BC \): Cable \( BC \) exerts a tensile force on \( C \). The direction of this force is along the cable \( BC \), pulling \( C \) towards \( B \). The angle of this cable with the horizontal (or the dashed line) is given as \( \theta \).
• Force from cable \( CD \): Cable \( CD \) exerts a tensile force on \( C \). The direction of this force is along the cable \( CD \), pulling \( C \) towards \( D \) (since \( D \) is attached to a weight, the force from \( CD \) on \( C \) is due to the weight hanging from \( D \), so it acts downward along \( CD \)).
• Force from cable \( CE \): Cable \( CE \) exerts a tensile force on \( C \). The direction of this force is along the cable \( CE \), pulling \( C \) towards \( E \) (since \( E \) is a support on the ground, the force from \( CE \) on \( C \) is upward along \( CE \) to balance the downward force from \( CD \) and other forces). Also, we need to consider the weight of the object attached to \( D \), but since we are drawing the free - body diagram for \( C \), the force from \( CD \) on \( C \) is equal to the weight of the object (by Newton's third law, if the object is in equilibrium, the tension in \( CD \) is equal to the weight of the object).

Step 3: Draw the forces on the free - body diagram of \( C \)

• Draw the circle for \( C \).
• Draw an arrow along \( AC \) towards \( A \) to represent the tension force \( \vec{T}_{AC} \).
• Draw an arrow along \( BC \) towards \( B \) to represent the tension force \( \vec{T}_{BC} \). The angle of this arrow with the horizontal (dashed line) is \( \theta \).
• Draw an arrow along \( CD \) downwards to represent the tension force \( \vec{T}_{CD} \) (equal to the weight \( W \) of the object at \( D \), so \( \vec{T}_{CD}=-W\hat{j} \) if we take upward as the positive \( y \) - direction).
• Draw an arrow along \( CE \) upwards to represent the tension force \( \vec{T}_{CE} \).

If we assume the weight of the object at \( D \) is \( W \), and we want to write the equilibrium equations (for a more detailed analysis), we can use the following:

In the \( x \) - direction (horizontal):
\( \sum F_{x}=T_{AC}\cos(0) - T_{BC}\cos\theta=0 \) (assuming \( AC \) is horizontal, if not, we need to adjust the angle of \( T_{AC} \) with the \( x \) - axis)

In the \( y \) - direction (vertical):
\( \sum F_{y}=T_{CE}-T_{CD}-T_{BC}\sin\theta = 0 \) (since \( T_{CD} = W \), we have \( T_{CE}=W + T_{BC}\sin\theta \))

But for the free - body diagram construction, the key is to represent all the forces (tensions from the cables) acting on particle \( C \) with their correct directions.

If we are just asked to construct the free - body diagram (the graphical representation), the steps are:

1. Represent particle \( C \) as a point (or a small circle).
2. Draw vectors for each force:

• Tension from \( AC \): direction along \( AC \) towards \( A \).
• Tension from \( BC \): direction along \( BC \) towards \( B \) at angle \( \theta \) from the horizontal.
• Tension from \( CD \): direction along \( CD \) downwards (towards \( D \)).
• Tension from \( CE \): direction along \( CE \) upwards (towards \( E \)).

Since the problem seems to be about constructing a free - body diagram (a common task in Engineering, which is a sub - field of Natural Science), and if we assume that we need to list the forces for the free - body diagram of \( C \), the forces are:

• Tension force from cable \( AC \) ( \( \vec{T}_{AC} \) )
• Tension force from cable \( BC \) ( \( \vec{T}_{BC} \) )
• Tension force from cable \( CD \) ( \( \vec{T}_{CD}=W\) downward, where \( W \) is the weight of the object at \( D \))
• Tension force from cable \( CE \) ( \( \vec{T}_{CE} \) upward)

If the question is to construct the free - body diagram (graphically), the answer would be the diagram with particle \( C \) and the four force vectors as described above.