Question

It is harder to move a door if you lean against it (along the plane of the door) toward the hinge than if you lean against the door perpendicular to its plane. Why is this so?

Short answer

** Answer: A door gets harder to move when a person leans against it along its plane towards the hinge because the torque generated in this case is zero, resulting in no rotational motion. On the other hand, when a person leans perpendicular to the door's plane, a non-zero torque is generated, which depends on the applied force and distance from the hinge, causing the door to move more easily.

Step-by-step solution

Analyze forces acting on the door when leaning along the plane of the door

When a person leans along the plane of the door towards the hinge, the force they apply on the door is parallel to the door itself. We can represent this force as \(F_{parallel}\).

Compute torque generated in the first case

Torque can be calculated as the cross product of force and distance vectors. In our case, the distance vector (\(r\)) is the perpendicular distance from the hinge to the point where force is applied on the door. The angle between \(F_{parallel}\) and \(r\) is 0 degrees. So, the torque generated (\(τ\)) can be calculated as: \(τ = r × F_{parallel} × \sin{0°}\) Since \(sin(0°) = 0\), the torque in this case is zero.

Analyze forces acting on the door when leaning perpendicular to the door

When a person leans against the door perpendicular to its plane, the force they apply on the door is perpendicular to the door itself. We can represent this force as \(F_{perpendicular}\).

Compute torque generated in the second case

Similar to step 2, we can calculate the torque generated in this case. The angle between \(F_{perpendicular}\) and \(r\) is 90 degrees. So, the torque generated (\(τ\)) can be calculated as: \(τ = r × F_{perpendicular} × \sin{90°}\) Since \(sin(90°) = 1\), the torque in this case is: \(τ = r × F_{perpendicular}\)

Compare the two cases

As we can see from the calculations, in the first case (leaning along the plane of the door), the torque generated is zero, while in the second case (leaning perpendicular to the door), the torque is non-zero and depends on the applied force and distance from the hinge. Thus, it becomes harder to move the door when leaning along its plane towards the hinge, as there's no torque generated to cause a rotational motion.

Key concepts

Rotational Motion

Rotational motion describes the movement of objects as they rotate around a central point or axis. This type of motion is an integral part of everyday mechanisms, from the revolving of wheels to the turning of a door knob. An object in rotational motion often adheres to Newton's laws, much like linear motion, but with a few twists specific to rotation, such as the concept of angular velocity, angular acceleration, and moment of inertia.

In the context of our example with the door, when force is applied at a point away from the hinge and perpendicular to the door, this induces a rotational motion. The door rotates around the hinge due to this applied force. Consider a situation where a person pushes the edge of the door: the further from the hinge the force is applied, the easier it will be to open the door – this is because the rotational effect of the force, or torque, is greater at larger distances from the axis of rotation.

Cross Product of Vectors

In physics and mathematics, the cross product of vectors is an operation that takes two vectors and produces a third vector that is perpendicular to the plane containing the first two. The magnitude of the resultant vector is equal to the area of the parallelogram that the vectors span, which is also a measure of how much the vectors 'point' in different directions.

For our door example, the cross product is essential in calculating the torque. The force vector and the distance vector from the hinge to the point where the force is applied create a plane, and the torque vector lies perpendicular to this plane. The effectiveness of that force to produce rotational motion, or torque, is directly related to the sine of the angle between the force and the distance vector. When the force is parallel to the door (and therefore along the same line as the distance vector), the cross product, and thus the torque, is zero, as the force is not causing any rotation.

Torque Calculation

Torque, also known as the moment of force, measures the tendency of a force to rotate an object around an axis, pivot, or hinge. The calculation of torque is central in understanding rotational motion in physics. Torque is the cross product of the position vector (the distance from the point of rotation to the point of force application) and the force vector, mathematically expressed as \(\tau = r \times F \times \sin\theta\), where \(\tau\) is the torque, \(r\) is the distance vector, \(F\) is the force applied, and \(\theta\) is the angle between the force and the distance vector.

In our door scenario, when the force is applied perpendicular to the door, the \(\sin\theta\) term equates to 1 since \(\theta = 90^{\circ}\), resulting in a maximum torque. This is why it is easier to open a door by pushing at the edge far from the hinge, as opposed to near it, where the distance \(r\) would be small, leading to a smaller torque.

Forces Acting on Rigid Bodies

A rigid body is an idealization of a solid body in which deformation is neglected. In the real world, every solid body has some deformation, but the assumption of rigidity is helpful to simplify the analysis of the mechanics involved. The forces acting on rigid bodies can produce two effects: translational motion and rotational motion. When all the forces and torques are considered, we can predict and understand how a rigid body will behave under those forces and torques.

Applying this to our door example, when force is applied along the plane of the door towards the hinge, it acts as a translational force, and since the door cannot move in that direction due to the hinge constraint, no rotational motion occurs. In contrast, when the force is applied perpendicular to the door, it is not aligned with any direct constraint and is able to produce a rotational motion. Understanding forces on rigid bodies helps us grasp why levers, wheels, and indeed doors, behave the way they do under various forces.