Member AB is supported by a cable BC and at A by a square rod
which fits loosely through the square hole at the end joint of the
member as shown. Determine the components of reaction at A and the
tension in the cable needed to hold the 800-lb cylinder in
equilibrium.
Answer
General guidance
Vector form of representation of location of a point from a reference point is called position vector.
A body is in equilibrium if vector sum of all the forces is equal to zero or moment of all force vectors about any point is equal to zero.
General sign convention for moment: The moment is considered positive in counter-clockwise direction and negative in clockwise direction.
Write the equilibrium equation of force vector.
Write the equilibrium equation of moment.
General sign convention for axis: Distance along the axis is positive and opposite to the axis is negative.
Step-by-step
Step 1 of 8
Draw the free body diagram of the member AB.
Tensions in supporting cables BC is represented as 
.
Since the joint 
can only translate along 
direction, its motion in all the remaining five (2 translational and 3 rotational) directions are arrested.
Consider
, 
are the reaction forces at 
due to the restriction of the rod to move in either of 
or 
directions.
Consider the moments acting at point 
along
,
, and 
directions to be
, 
, and 
respectively.
Step 2 of 8
Calculate the length of the cable by using the following relation:
Here, 
, 
, and 
are the length of 
, 
, and 
.
Substitute 
for 
, 
for 
, and 
for 
.
Using trigonometric relation length of the cable is calculated.
Step 3 of 8
Write the force equilibrium equation in the 
direction.
Substitute 
for 
.
Since tension in cable BC makes an angle with x-axis,
.
Therefore, 
Write the force equilibrium equation in the 
direction.
Therefore, the y-component of reaction at A is 
.
Using force equilibrium in y-direction component of reaction at A is calculated.
Step 4 of 8
Write the force equilibrium equation in the 
direction.
Here, 
is the weight of the cylinder acting downwards.
Substitute 
for 
and 0 for 
.
Therefore, the z-component of reaction at A is 
.
Using force equilibrium in z-direction component of reaction at A is calculated.
Step 5 of 8
Consider the moment equilibrium in x direction.
Substitute for and 
for 
.
Therefore, the x-component of moment at A is 
.
Using moment equilibrium in x-direction component of reaction at A is calculated.
Step 6 of 8
Consider the moment equilibrium in y direction.
Therefore, the y-component of moment at A is 
.
Using moment equilibrium in y-direction component of moment at A is calculated.
Step 7 of 8
Consider the moment equilibrium in 
direction.
Therefore, the z-component of moment at A is .
Using moment equilibrium in z-direction component of moment at A is calculated.
Step 8 of 8
Write the force equilibrium equation in the 
direction.
Therefore, the tension in the cable 
is 
.
Tension in cable BC is calculated by applying force balance in x-direction.
Answer
Therefore, the y-component of reaction at A is .
Therefore, the z-component of reaction at A is .
Therefore, the x-component of moment at A is .
Therefore, the y-component of moment at A is .
Therefore, the z-component of moment at A is .
Therefore, the tension in the cable is .
Answer only
Therefore, the y-component of reaction at A is .
Therefore, the z-component of reaction at A is .
Therefore, the x-component of moment at A is .
Therefore, the y-component of moment at A is .
Therefore, the z-component of moment at A is .
Therefore, the tension in the cable is .
