Question

A door, essentially a uniform rectangle of height \(2.00 \mathrm{~m}\) width \(0.80 \mathrm{~m}\), and weight \(100.0 \mathrm{~N}\), is supported at one edge by two hinges, one \(30.0 \mathrm{~cm}\) above the bottom of the door and one \(170.0 \mathrm{~cm}\) above the bottom of the door. Calculate the horizontal components of the forces on the two hinges .

Short answer

Answer: The horizontal components of the forces on the two hinges are both 58.82 N.

Step-by-step solution

Identify forces acting on the door

First, we need to identify all the forces acting on the door. There are three main forces: 1. The gravitational force acting downward, which is equivalent to the weight of the door, at its center of mass. 2. The force exerted by the upper hinge, which has both horizontal and vertical components. 3. The force exerted by the lower hinge, which also has horizontal and vertical components.

Apply the equilibrium equations

Since the door is in equilibrium, both the net force and the net torque acting on it must be zero. Therefore, we have three equilibrium equations: 1. The sum of the forces in the vertical direction is zero: \(F_{hy1} + F_{hy2} = 100 \mathrm{~N}\), 2. The sum of the forces in the horizontal direction is zero: \(F_{hx1} - F_{hx2} = 0\), 3. The net torque about any point is zero. We can choose any arbitrary point to analyze the torque, and it is usually easiest to use the point where one of the hinges is located in order to simplify calculations. For this solution, we will use the lower hinge as the point of analysis: \((1.70\ \mathrm{m})(F_{hx2}) - (1.00\ \mathrm{m})(100 \mathrm{~N}) = 0\).

Solve the equilibrium equations

Now we will solve the equilibrium equations for the horizontal components of the forces exerted by the two hinges. From the second equation, we can see that the horizontal components of the forces on the two hinges must be equal in magnitude: \(F_{hx1} = F_{hx2}\). We can then use the third equation to find the value of \(F_{hx2}\): \((1.70\ \mathrm{m})(F_{hx2}) - (1.00\ \mathrm{m})(100 \mathrm{~N}) = 0 \Rightarrow F_{hx2} = \frac{100\ \mathrm{N}}{1.70\ \mathrm{m}} = 58.82\ \mathrm{N}\) Since \(F_{hx1} = F_{hx2}\), we have: \(F_{hx1} = 58.82\ \mathrm{N}\).

Final answer

The horizontal components of the forces on the two hinges are: \(F_{hx1} = 58.82\ \mathrm{N}\), \(F_{hx2} = 58.82\ \mathrm{N}\). These are the forces that the two hinges must exert in the horizontal direction to keep the door in equilibrium.

Key concepts

Torque and Equilibrium

When addressing physics equilibrium problems, it's instrumental to comprehend the principle of torque and its relation to equilibrium. Torque, often referred to as the moment of force, is a measure of the rotational effect produced when a force is applied at a distance from a pivot point. The concept can be visualized by thinking about opening a door; you apply a force to the handle, which is at a certain distance from the hinges (the pivot), creating a turning effect.

For an object to be in equilibrium, specifically rotational equilibrium, the sum of all torques on the object must be zero. This condition assures that the object does not rotate and maintains a steady orientation. When solving a problem involving torque, it's important to keep in mind that the direction in which the force is exerted matters as much as the magnitude of the force and its distance from the pivot.

Forces in Equilibrium

The balance of forces is crucial in understanding equilibrium in physics problems. When we say that forces are in equilibrium, we mean that an object is either at rest or moving at a constant speed in a straight line. This happens because the net force acting upon the object is zero. In our door example, the forces exerted by the hinges provide the counterbalance to the weight of the door.

When encountering a problem where forces are in equilibrium, it is helpful to draw a free body diagram to visualize the forces. Since these forces need to cancel each other out, for the door to be in equilibrium, the vertical forces (the weight of the door and the vertical components of forces from the hinges) and the horizontal forces (the horizontal components of forces from the hinges) must sum to zero. This is how we derive the equations used to solve for the unknown forces.

Static Equilibrium

Static equilibrium is a specific case of equilibrium where the object is at rest and not moving. In static equilibrium, not only must the net force be zero, but the net torque must also be zero. The door in our exercise is a perfect example; it's not moving which means it's in static equilibrium.

To analyze a static equilibrium situation, we use the conditions of equilibrium: that the sum of the forces in all directions (both horizontal and vertical) is zero, and that the net torque about any axis is zero. By setting up equations for these conditions, as we did in the solution for the door exercise, we can solve for unknown quantities. When the door's weight is distributed evenly, as in a uniform rectangle, its center of mass is at its geometric center, and that is where the gravitational force acts.