Ch 11: Equilibrium & Elasticity

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 11: Equilibrium & Elasticity

Problem 17c

Chapter 11, Problem 17c

A 9.00 m-long uniform beam is hinged to a vertical wall and held horizontally by a 5.00 m-long cable attached to the wall 4.00 m above the hinge (Fig. E11.17). The metal of this cable has a test strength of 1.00 kN, which means that it will break if the tension in it exceeds that amount. Find the horizontal and vertical components of the force the hinge exerts on the beam. Is the vertical component upward or downward?

Verified step by step guidance

Identify the forces acting on the beam: the tension in the cable, the weight of the beam, and the reaction forces at the hinge (horizontal and vertical components).

Calculate the angle θ between the cable and the horizontal using trigonometry: \( \theta = \arctan\left(\frac{4.0}{5.0}\right) \).

Determine the tension in the cable. Since the cable's test strength is 1.00 kN, assume the tension is at this maximum value for safety calculations.

Resolve the tension into horizontal and vertical components: \( T_x = T \cdot \cos(\theta) \) and \( T_y = T \cdot \sin(\theta) \).

Apply the conditions for equilibrium: sum of horizontal forces is zero, sum of vertical forces is zero, and sum of torques about the hinge is zero. Use these to solve for the horizontal and vertical components of the hinge force.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Torque and Equilibrium

Torque is the rotational equivalent of force, calculated as the product of force and the perpendicular distance from the pivot point. For an object in equilibrium, the sum of all torques acting on it must be zero. In this problem, the beam is in static equilibrium, meaning the torques due to the weight of the beam and the tension in the cable must balance each other around the hinge.

Components of Force

Forces can be broken down into horizontal and vertical components, which are useful for analyzing the effects of forces in different directions. In this scenario, the hinge exerts a force on the beam that can be decomposed into horizontal and vertical components. These components are crucial for determining the net force and ensuring the beam remains in equilibrium.

Tension in the Cable

Tension is the force exerted by a cable or rope when it is pulled tight by forces acting from opposite ends. The tension in the cable must not exceed its test strength of 1.00 kN to prevent breaking. The tension contributes to the equilibrium of the beam by providing a counteracting force to the weight of the beam, and it affects the horizontal and vertical forces at the hinge.

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Related Practice

Textbook Question

Suppose that you can lift no more than 650 N (around 150 lb) unaided.

(a) How much can you lift using a 1.4-m-long wheelbarrow that weighs 80.0 N and whose center of gravity is 0.50 m from the center of the wheel (Fig. E11.16)? The cen-ter of gravity of the load car-ried in the wheelbarrow is also 0.50 m from the center of the wheel. (b) Where does the force come from to enable you to lift more than 650 N using the wheelbarrow?

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Textbook Question

A 9.00-m-long uniform beam is hinged to a vertical wall and held horizontally by a 5.00-m-long cable attached to the wall 4.00 m above the hinge (Fig. E11.17). The metal of this cable has a test strength of 1.00 kN, which means that it will break if the tension in it exceeds that amount. What is the heaviest beam that the cable can support in this configuration?

A 15,000-N crane pivots around a friction-free axle at its base and is supported by a cable making a 25° angle with the crane (Fig. E11.18). The crane is 16 m long and is not uniform, its center of gravity being 7.0 m from the axle as measured along the crane. The cable is attached 3.0 m from the upper end of the crane. When the crane is raised to 55° above the horizontal holding an 11,000-N pallet of bricks by a 2.2-m, very light cord, find the tension in the cable. Start with a free-body diagram of the crane.

A nonuniform beam 4.50 m long and weighing 1.40 kN makes an angle of 25.0° below the horizontal. It is held in position by a frictionless pivot at its upper right end and by a cable 3.00 m farther down the beam and perpendicular to it (Fig. E11.20). The center of gravity of the beam is 2.00 m down the beam from the pivot. Lighting equipment exerts a 5.00-kN downward force on the lower left end of the beam. Find the tension T in the cable and the horizontal and vertical components of the force exerted on the beam by the pivot. Start by sketching a free-body diagram of the beam.

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Textbook Question

A circular steel wire 2.00 m long must stretch no more than 0.25 cm when a tensile force of 700 N is applied to each end of the wire. What minimum diameter is required for the wire?

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Textbook Question

Suppose that you can lift no more than 650 N (around 150 lb) unaided.

How much can you lift using a 1.4 m-long wheelbarrow that weighs 80.0 N and whose center of gravity is 0.50 m from the center of the wheel (Fig. E11.16)? The center of gravity of the load carried in the wheelbarrow is also 0.50 m from the center of the wheel.

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