Question

A uniform, 7.5-m-long beam weighing 6490 N is hinged to a wall and supported by a thin cable attached 1.5 m from the free end of the beam. The cable runs between the beam and the wall and makes a 40\(^\circ\) angle with the beam. What is the tension in the cable when the beam is at an angle of 30\(^\circ\) above the horizontal?

Short answer

The tension in the cable is approximately 4430 N.

Step-by-step solution

Identify the Forces Acting on the Beam

The beam experiences three main forces: the gravitational force (weight of the beam), the tension in the cable, and the reaction force at the hinge. The weight of the beam acts downward at its center of gravity (midpoint of the beam, at 3.75 m from the hinge). The tension in the cable acts along the cable direction, and the reaction force at the hinge supports the beam against these forces.

Determine the Conditions for Equilibrium

For the beam to be in static equilibrium, the sum of the forces and the sum of the moments around any point must be zero. Choose the point at the hinge to calculate torques, as this eliminates the reaction force at the hinge from the moment equation.

Calculate the Torque Due to Beam’s Weight

Calculate the torque due to the weight of the beam relative to the hinge. The weight acts at the midpoint (3.75 m from the hinge). Torque due to weight is given by:\[ \tau_{weight} = F_{weight} \cdot d \cdot \cos(\theta) \]where \( F_{weight} = 6490 \text{ N} \), \( d = 3.75 \text{ m} \), and \( \theta = 30^\circ \). Thus,\[ \tau_{weight} = 6490 \cdot 3.75 \cdot \cos(30^\circ) \].

Calculate the Torque Due to Tension in the Cable

The torque due to tension acts at the point where the cable is attached, 1.5 m from the beam's free end, hence 6 m from the hinge. The angle between the tension and the beam is \( 40^\circ \), so:\[ \tau_{tension} = T \cdot 6 \cdot \sin(40^\circ) \]where \( T \) is the tension in the cable.

Set up the Torque Balance Equation

Set the sum of the torques around the hinge to zero:\[ \tau_{weight} = \tau_{tension} \]Substitute the expressions from Steps 3 and 4:\[ 6490 \cdot 3.75 \cdot \cos(30^\circ) = T \cdot 6 \cdot \sin(40^\circ) \]

Solve for the Tension in the Cable

Solve the equation from Step 5 for \( T \):\[ T = \frac{6490 \cdot 3.75 \cdot \cos(30^\circ)}{6 \cdot \sin(40^\circ)} \]Calculate \( T \) to find the tension in the cable.

Key concepts

Torque Calculations

Torque is essentially the rotational equivalent of linear force. Imagine opening a door. The force you apply with your hand is more effective at rotating the door if you push it near the handle rather than close to its hinges. This is because torque depends not only on the force applied but also on the distance from the pivot point, called the "lever arm."

The formula for calculating torque is \( \tau = F \cdot d \cdot \sin(\theta) \), where \( \tau \) is the torque, \( F \) is the force applied, \( d \) is the distance from the pivot, and \( \theta \) is the angle between the force and the lever arm. In our beam example, the weight of the beam and the tension in the cable each create a torque around the hinge. The sum of these torques determines the beam's balance.

• Torque due to weight: Acts through the beam's center of gravity.
• Torque due to tension: Acts at the point of cable attachment.

When these torques are precisely balanced, the beam remains in equilibrium.

Statics in Physics

Statics is a branch of mechanics that deals with objects at rest or in a state of constant velocity. The key idea in statics is that an object remains stationary because the sum of forces and the sum of torques acting on it are zero.

In the case of the beam attached to the wall, the equilibrium condition involves:

• Sum of forces: The total forces acting in any direction must be zero. This means vertical forces like gravity and the tension's vertical component must cancel each other.
• Sum of torques: The total rotational forces around a pivot point must be zero. This ensures that the beam doesn't start to rotate.

In our exercise, choosing the hinge as the pivot point eliminates the reaction force from the torque calculations. This simplifies determining the beam's equilibrium state.

Tension in Cables

The tension in a cable is the force exerted along its length, counteracting any applied forces that it supports. In physics problems involving beams, cables play a crucial role in maintaining stability and balance.

Consider the cable attached to the beam in our exercise. This cable supports the beam against gravity and prevents it from falling by creating tension. This tension can be calculated using the known weights, angles, and distances. The tension acts along the line of the cable and can be decomposed into components using trigonometry:

• Horizontal component: Contributes to offsetting any horizontal forces, though not directly relevant in this problem.
• Vertical component: Counters the downward force of gravity.

By forming an angle with the beam, the tension provides just enough force to keep the system in perfect balance, illustrating how even a simple thin cable can effectively stabilize a heavy structure like a beam.