Question

A horizontal bar that is \(2.261 \mathrm{~m}\) long and has a mass of \(82.45 \mathrm{~kg}\) is hinged to a wall. The bar is supported at its other end by a cable attached to the wall. The cable has a tension of \(618.8 \mathrm{~N}\). What is the angle between the cable and the bar?

Short answer

Answer: The angle between the cable and the horizontal bar is approximately 39.8°.

Step-by-step solution

Identify the forces acting on the bar

There are three forces acting on the bar: the gravitational force acting at the center of mass of the bar, the tension force acting at the point where the cable is connected, and the reaction force at the hinge. We will call these forces F_gravity, F_tension, and F_reaction, respectively.

Calculate the gravitational force acting on the bar

The gravitational force acting on the bar can be found using the formula F_gravity = m * g, where m is the mass of the bar and g is the acceleration due to gravity (approximately 9.81 m/s²). In our problem, m = 82.45 kg, so: F_gravity = 82.45 kg * 9.81 m/s² ≈ 808.25 N

Set up the torque equation

As the bar is in equilibrium, the sum of the torques acting on it must be zero. We will use the hinged point as the reference point to calculate the torques. The torque due to the gravitational force is clockwise (positive), while the torque due to the tension in the cable is counter-clockwise (negative). Thus, the equation is: torque(F_gravity) + torque(F_tension) = 0

Calculate the torque due to the gravitational force

The torque due to the gravitational force can be calculated using the formula torque(F_gravity) = F_gravity * d, where d is the distance from the center of mass of the bar to the hinge. Since the bar is uniform, the center of mass is at half of its length, so d = 2.261 m / 2 = 1.1305 m. Thus, torque(F_gravity) = 808.25 N * 1.1305 m ≈ 913.51 Nm (clockwise)

Calculate the torque due to the tension in the cable

The torque due to the tension in the cable can be calculated using the formula torque(F_tension) = F_tension * L * sin(theta), where L is the length of the bar and theta is the angle between the cable and the bar. In our problem, F_tension = 618.8 N and L = 2.261 m. Since the sum of torques is zero: F_tension * L * sin(theta) = 913.51 Nm (clockwise) 618.8 N * 2.261 m * sin(theta) = 913.51 Nm

Solve for the angle(theta)

Now, we need to solve for theta in the equation: sin(theta) = 913.51 Nm / (618.8 N * 2.261 m) sin(theta) ≈ 0.6366 theta = arcsin(0.6366) theta ≈ 39.8° Hence, the angle between the cable and the bar is approximately 39.8°.

Key concepts

Gravitational Force

Understanding the concept of gravitational force is crucial when analyzing the balance of forces in physics problems. The force of gravity is what keeps our feet planted on the ground, and in the context of our exercise, it's what causes the bar to pull downward.

Every object with mass experiences gravitational force, which is proportional to its mass and the acceleration due to gravity (\( g \)). This force is calculated using the formula \( F_{\text{gravity}} = m \times g \), where \( m \) is mass and \( g \) is approximately \( 9.81 \text{m/s}^2 \) on Earth. In our example, the bar's mass is \( 82.45 \text{kg} \) and the gravitational force is \( 808.25 \text{N} \) acting at the center of the bar's mass.

Gravitational force is a key player in various phenomena, and it's essential for students to recognize its role in equilibrium scenarios. Apart from its role in this problem, it dictates planetary orbits and influences the tides in our oceans.

Torque Calculation

Torque is a measure of how much a force acting on an object causes that object to rotate. In the problem at hand, we are particularly interested in how torque affects the bar's rotational equilibrium. To say that an object is in rotational equilibrium, the total torque acting on it must be zero.

The calculation of torque (\( \tau \)) is given by the formula \( \tau = F \times d \times \sin(\theta) \), where \( F \) is the force applied, \( d \) is the distance from the pivot point to the point where the force is applied, and \( \theta \) is the angle between the force vector and the lever arm.

In our example, there are two main torques to consider: the gravitational torque which is acting in a clockwise direction and calculated as the product of the gravitational force and the distance from the hinge to the center of mass, and the torque due to the cable tension which acts counter-clockwise. Their sum needs to be zero for the bar to be in equilibrium, and this condition is crucial for solving the problem.

Tension in Cables

Tension is the force transmitted through a string, cable, or chain when it is pulled tight by forces acting from opposite ends. In our exercise, the tension in the cable is supporting the weight of the bar and preventing it from rotating around the hinge. The tension force is always directed along the cable and towards the object to which the cable is attached.

In physics problems, tension is often the force we need to find or the given value we use to calculate other unknowns. In this scenario, the tension in the cable is a known value of \( 618.8 \text{N} \). However, instead of acting straight upwards, it applies a force at an angle, which introduces a torque component.

Understanding how the tension contributes to the torque allows students to see the relationship between linear forces (like tension) and rotational motion (torque). This knowledge is not only applicable in theoretical physics but also in practical engineering situations such as the design of bridges and cable cars.