Question

A trapdoor on a stage has a mass of \(19.2 \mathrm{~kg}\) and a width of \(1.50 \mathrm{~m}\) (hinge side to handle side). The door can be treated as having uniform thickness and density. A small handle on the door is \(1.41 \mathrm{~m}\) away from the hinge side. A rope is tied to the handle and used to raise the door. At one instant, the rope is horizontal, and the trapdoor has been partly opened so that the handle is \(1.13 \mathrm{~m}\) above the floor. What is the tension, \(T,\) in the rope at this time?

Short answer

Answer: The tension in the rope is approximately 99.91 N.

Step-by-step solution

Find the gravitational force acting on the trapdoor

Let's first find the gravitational force acting on the trapdoor. The gravitational force acting on an object is given by the equation: \(F_g = m \times g\) where \(F_g\) is the gravitational force, \(m\) is the mass of the object, and \(g\) is the acceleration due to gravity (approximately 9.81 m/s²). Given that the mass of the trapdoor is 19.2 kg, we can calculate the gravitational force: \(F_g = 19.2 \times 9.81 = 188.352 \ \text{N}\) The gravitational force on the trapdoor is 188.352 N.

Find the center of mass of the trapdoor

The trapdoor is a uniform rectangle, so the center of mass is at the midpoint of the rectangle's width. The width of the trapdoor is 1.50 m, so the center of mass is located at: \(x_{cm} = \frac{1}{2} \times 1.50 = 0.75 \ \text{m}\) from the hinge side of the door.

Calculate the lever arms for the gravitational force and the tension force

The lever arm (or moment arm) is the perpendicular distance from the pivot point (in this case, the hinge side) to the line of action of the force. Let's find the lever arms of the gravitational force and the tension force. For the gravitational force, its line of action passes horizontally through the center of mass. To find the lever arm, we consider the perpendicular distance from the hinge side to the center of mass: \(L_{F_g} = x_{cm} = 0.75 \ \text{m}\) For the tension force, the line of action passes horizontally through the handle. The lever arm will be the perpendicular distance from the hinge side to the handle, which is given as 1.41 m: \(L_{T} = 1.41 \ \text{m}\)

Analyze the torques acting on the trapdoor

Now that we know the gravitational force, the center of mass, and the lever arms, we can analyze the torques acting on the trapdoor. The net torque on the trapdoor is zero when it is in equilibrium, which is given by the following equation: \(\sum{\tau} = F_g L_{F_g} - T L_T = 0\) We can rearrange this equation to solve for tension, T: \(T = \frac{F_g L_{F_g}}{L_T}\)

Calculate the tension in the rope

Now we can plug in the gravitational force, the lever arms, and the center of mass to find the tension in the rope: \(T = \frac{(188.352 \ \text{N})(0.75 \ \text{m})}{1.41 \ \text{m}}\) \(T = 99.91 \ \text{N}\) The tension in the rope at this instant is approximately 99.91 N.

Key concepts

Gravitational Force

When we're dealing with objects on or near the Earth's surface, the gravitational force is a key player. It's the force that pulls everything towards the Earth's center, and it's measured in newtons (N). In physics problems like the trapdoor example, we calculate gravitational force using the formula \(F_g = m \times g\), where \(m\) represents the mass of the object and \(g\) is the acceleration due to gravity, which is approximately \(9.81 \text{m/s}^2\) on Earth.

Understanding the role of gravitational force is critical when analyzing forces and torques, as it's this force that needs to be balanced when we're considering mechanical equilibrium – a state where an object is at rest or moves with constant velocity, meaning there's no unbalanced force acting upon it.

Center of Mass

The center of mass is essentially the point where an object's mass is evenly distributed around in all directions. For symmetric objects with uniform density, like the trapdoor in our problem, this point is at a geometric center - in this case, at the halfway point of its width. Knowing this location allows us to understand how gravitational force acts on the object and is crucial for calculating torques, as the force acts as if it were concentrated at this single point.

Identifying the center of mass helps us simplify complex objects into more manageable concepts by treating the entire mass as if it's focused at one spot when we're dealing with linear forces and rotational effects.

Torque Equilibrium

Torque equilibrium is a fundamental concept when it comes to rotational motion. It states that if an object is not spinning uncontrollably or accelerating in its rotation, the net torque must be zero. In other words, the torques (which are the rotational equivalents of forces) causing clockwise rotation must be balanced by the torques causing counterclockwise rotation.

In the trapdoor scenario, we apply this principle by acknowledging that the tension in the rope creates a torque to lift the door open, while the gravitational force causes a torque trying to close it. Since the door is partly open without any angular acceleration, the torques must be in equilibrium - they cancel out, leading us to the equation \(\sum{\tau} = F_g L_{F_g} - T L_T = 0\). This simple equation is powerful as it helps us find unknown quantities like the tension in the rope.

Lever Arm

The lever arm concept is pivotal in understanding how torques are generated. It is the distance from the axis of rotation to the point where the force is applied and it's perpendicular to the direction of the force. The lever arm is a measure of how effectively a force can cause an object to rotate.

In the context of the textbook problem, the trapdoor acts as a lever with the hinge side as the pivot point. From this pivot point, two lever arms play a role: one for gravitational force which acts at the center of mass and one for tension which acts where the rope is attached. It is the length of these lever arms that determines how much rotational force—or torque—each applied force generates. The careful manipulation of lever arms is essential in solving problems involving torque and rotational equilibrium.