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Fundamentals Of Physics
Found in: Page 346

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Short Answer

The system in Fig. 12-38 is in equilibrium. A concrete block of mass225 kg hangs from the end of the uniform strut of mass45.0 kg. A cable runs from the ground, over the top of the strut, and down to the block, holding the block in place. For anglesϕ = 30.0° andθ= 45.0° , find (a) the tension T in the cable and the (b) horizontal and (c) vertical components of the force on the strut from the hinge.

a)The tension in the cable is.6.63×103N

b)Horizontal component of the force is5.74×103N

c) Vertical component of the force is5.96×103N

See the step by step solution

Step by Step Solution

Step 1: Listing the given quantities

Theconcrete block of mass225 kg

Strut of mass45.0 kg

ϕ = 30.0°θ= 45.0°

Step 2: Understanding the concept of the horizontal and vertical componentof the force

In the free body diagram of the given situation, we notice the acting torques on the body both horizontally and vertically.Here, the acting torque results from the applied force along the length of the cable hanging from the strut and along the weight of the block to balance it at the strut.The vertical force acting on the body about the hinge is due to the tension resulting from the hanging to the strut, also the downward pull due to the weight of the block, and the downward pull due to the weight of the strut that acts at the center of the strut. Then using Newton's laws of motion, we balance the acting torques in an equation separately for both the horizontal and the vertical directions. Then, accordingly, calculate the required values as per the problem

Step 3: Calculation of the tension in the cable

We note that the angle between the cable and the strut is

α=θ-ϕ = 45°-30°=15°

The angle between the strut and any vertical force(like the weights in the problem) is

β= 90°-45°=45°

DenotingM= 225kgandm=45.0kgand as the length of the boom, we compute torques about the hinge and find


The unknown lengthcancels out and we obtainT= 6.63×103N

The tension in the cable is.6.63×103N

Step 4:  Calculation of thehorizontal component of the force


Since the cable is at30° from horizontal, then horizontal equilibrium of forces requires that the horizontal hinge force be


Horizontal component of the force is5.74×103N

Step 5: Calculation of thevertical component of the force


Vertical equilibrium of forces gives the vertical hinge force component


Vertical component of the force is5.96×103N

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