Question

A horizontal bar that is \(2.381 \mathrm{~m}\) long is hinged to a wall. The bar is supported at its other end by a cable attached to the wall, which makes an angle of \(\theta=42.75^{\circ}\) with the bar. The cable has a tension of \(599.3 \mathrm{~N}\). What is the mass of the bar?

Short answer

Answer: The mass of the horizontal bar is approximately 43.80 kg.

Step-by-step solution

Calculate the torque exerted by the cable

Calculate torque exerted by the cable. We will use the formula: $$ \tau = F \times d \times \sin(\theta) $$ Where: - \(\tau\) is the torque - \(F\) is the force (tension in the cable) - \(d\) is the distance from the hinge to the point where the force is applied - and \(\sin(\theta)\) is the angle between the force and the lever arm In our case, the tension (\(F\)) is \(599.3\,\text{N}\), the distance (\(d\)) is the length of the bar \(2.381\,\text{m}\), and the angle (\(\theta\)) is \(42.75^\circ\). First, find the sine of the angle: $$ \sin(42.75^\circ) = 0.688235 $$ Now, calculate the torque exerted by the cable: $$ \tau = 599.3\,\text{N} \times 2.381\,\text{m} \times 0.688235 $$ $$ \tau = 979.35\,\text{N}\text{m} $$ The torque exerted by the cable is \(979.35\,\text{N}\text{m}\).

Calculate the torque exerted by the weight of the bar

Next, we will calculate the torque exerted by the weight of the bar. The weight of the bar acts downward at the center of the bar, which is at a distance of half the length of the bar from the hinge (\(0.5 \times 2.381\,\text{m}=1.1905\,\text{m}\)). Let \(m\) be the mass of the bar and \(g = 9.81\,\text{m/s}^2\) be the acceleration due to gravity. Then the weight of the bar is \(mg\). The angle between the weight of the bar and the lever arm is \(90^\circ\). Therefore, the torque exerted by the weight of the bar is: $$ \tau_{\text{weight}} = mg \times 1.1905 \,\text{m} \times \sin(90^\circ) $$ As \(\sin(90^\circ) = 1\), the torque exerted by the weight of the bar can be written as: $$ \tau_{\text{weight}} = mg \times 1.1905\,\text{m} $$

Apply the condition of equilibrium

Since the bar is in equilibrium, the total torque acting on it must be zero. This means the torque exerted by the cable must be equal and opposite to the torque exerted by the weight of the bar: $$ \tau_{\text{weight}} = -\tau $$ Substituting the expressions from the previous steps: $$ mg \times 1.1905\,\text{m} = -979.35\,\text{N}\text{m} $$

Solve for the mass of the bar

Now, solve the equation for the mass (\(m\)) of the bar: $$ m = \frac{-979.35\,\text{N}\text{m}}{1.1905\,\text{m} \times 9.81\,\text{m/s}^2} $$ $$ m = 43.80\,\text{kg} $$ Therefore, the mass of the bar is approximately \(43.80\,\text{kg}\).

Key concepts

Torque Calculation

Torque, often represented as \( \tau \), measures how much a force acting on an object causes that object to rotate. The equation used to calculate torque is \( \tau = F \times d \times \sin(\theta) \), where \( F \) is the force applied, \( d \) is the distance from the rotation point to where the force is applied (also known as the lever arm), and \( \theta \) is the angle between the force vector and the lever arm.

In the context of the exercise, the tension in the cable creates a rotational effect on the bar, which is counteracted by the bar's weight. By computing the \( \sin(\theta) \) first and then multiplying by the force and lever arm, we find the torque exerted by the cable. Understanding this concept is crucial, as correctly calculating torque is a fundamental skill in statics and can help prevent costly engineering mistakes.

Forces in Equilibrium

When talking about forces in equilibrium, we refer to a condition known as 'static equilibrium'. This state occurs when all the forces and moments (torques) acting on a body are balanced, and there's no net motion. In simple terms, for every action, there is an equal and opposite reaction. The equilibrium condition for torque is \( \sum \tau = 0 \), meaning that the sum of all torques must be zero for the body to be in equilibrium.

Referring to our textbook problem, the horizontal bar remains stationary because the torque produced by the tension in the cable is equal in magnitude but opposite in direction to the torque due to the bar's weight. This balance allows us to set up an equation where the torque exerted by the cable's tension is equal to the torque due to the bar’s weight, enabling us to solve for the mass of the bar.

Statics

Statics is the branch of mechanics that deals with the analysis of loads (force, torque or momentum) on physical systems in static equilibrium, that is, in a state where the relative positions of the subsystems do not vary over time. When it comes to statics, understanding how to apply the principles of equilibrium is fundamental; these principles assert that a body at rest or moving with constant velocity has no net force acting on it and no net torque.

In the case of the bar and cable system in the problem, statics principles help us understand the forces at play and how they interact to keep the bar stationary. The condition of equilibrium is not just a mathematical statement but a reflection of physical balance in the real world. Knowing how to manipulate and apply the equations of statics is vital for solving real-world engineering problems and is an indispensable skill for students in engineering and physics.