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 .