Question

Consider Lewis and Tolman's lever paradox. The pivot B of a right-angled lever \(A B C\) is fixed at the origin of \(S^{\prime}\) while \(A\) and \(C\) lie on the positive \(x\) and \(y\)-axes, respectively, and \(\mathrm{AB}=\mathrm{BC}=a\). Two numerically equal forces of magnitude \(f\) act at \(A\) and \(C\), in directions \(B C\) and \(B A\), respectively, so that equilibrium obtains. Show that in the usual second frame S there is a clock wise couple \(f a v^{2} / c^{2}\) duc to these forces. Why does the lever not turn? Resolve the paradox along the lines of the preceding exercise. [Note: Intuitively it is not difficult to see how the angular momentum \(h\) of the lever continually increases in \(S\) in spite of its non-rotation. The force at \(C\) continually does positive work on the lever, while the reaction at B does equal negative work. Thus energy flows in at \(\mathrm{C}\) and out at B. But an energy current corresponds to a momentum density. So there is a constant nonmaterial momentum density along the limb \(\mathrm{BC}\) towards \(\mathrm{B}\). It is the continual increase of its moment about the origin of \(S\) that causes \(\mathrm{d} / \mathrm{d} t\) to be non- zero.]

Short answer

Answer: The constant angular momentum increase in frame S is caused by the non-material momentum density along limb BC towards point B. In frame S, the force at point C continuously does positive work on the lever while the reaction at B does equal negative work, resulting in energy flowing in at point C and out at point B.

Step-by-step solution

Understanding the lever paradox

In the given problem, the lever ABC is fixed at the origin of frame S' with a right angle formed at point B. Point A lies on the positive x-axis, and point C lies on the positive y-axis. Given that AB and BC have length a, and two force magnitudes, f, act at A and C in the direction of BC and BA, respectively. In this scenario, the lever is in equilibrium.

Calculate the clockwise couple

In frame S, the lever's ends A and C are contracted by the length contraction factor, given by \(\sqrt{1 - (v^2/c^2)}\). The coordinates of point A in frame S are \((a\sqrt{1 - (v^2/c^2)}, 0)\) and the coordinates of point C in frame S are \((0, a\sqrt{1 - (v^2/c^2)})\). We can calculate the moment of force or torque about point B by using the perpendicular distance between the line of action of a force and the point B. For forces acting through points A and C, we have: Torque_A = Force_A * Perpendicular_distance_AB = f * a * (1 - (v^2/c^2)) Torque_C = Force_C * Perpendicular_distance_BC = f * a * (1 - (v^2/c^2)) The clockwise couple in frame S is given by: Couple_S = Torque_A + Torque_C = 2 * f * a * (1 - (v^2/c^2)) = f * a * v^2 / c^2

Explain the non-rotation and resolve the paradox

Although there would be a clockwise couple due to both forces in frame S, the lever does not turn. This apparent paradox is resolved by understanding that in frame S, there is a momentum density along limb BC towards point B. The force at point C continuously does positive work on the lever while the reaction at B does equal negative work. As a result, energy flows in at point C and out at point B. This non-material momentum density is responsible for the constant angular momentum increase in frame S, even though the lever does not rotate. In frame S', the lever is in equilibrium and does not rotate, so there is no unresolved paradox in this situation.

Key concepts

Special Relativity

Special relativity is a foundational theory in physics developed by Albert Einstein that provides a way to describe the behavior of objects at high speeds—close to the speed of light. When objects move at a significant fraction of the speed of light, the usual Newtonian mechanics no longer apply, and the behaviors predicted by relativity take over. Some of the key principles of special relativity include the constancy of the speed of light in a vacuum for all observers regardless of their relative motion, and the fact that the laws of physics are the same in all inertial frames of reference. One of the consequences of these principles is time dilation, where a moving clock ticks slower than a stationary clock, and length contraction, which asserts that objects physically shorten in the direction of motion relative to an observer.

When applied to the lever paradox, special relativity provides a framework for understanding how forces and motions behave differently than they would in a purely Newtonian context, especially when the relative motion at high velocities leads to surprising consequences such as non-material momentum density along a lever arm.

Length Contraction

Length contraction is a phenomenon in special relativity where the length of an object moving at a significant fraction of the speed of light appears to be shorter when measured by an observer at rest relative to the object. It occurs in the direction of motion and becomes more pronounced as the object's velocity approaches the speed of light. This counterintuitive effect is a geometric consequence of space-time behaving differently from our everyday experiences.

For example, in the context of the lever paradox, as the lever moves relative to an observer in frame S, segments AB and BC contract. This contraction affects the calculation of torque and ultimately the couple on the lever, which relies on the perpendicular distance between the force and the pivot point. Understanding this length contraction is key to working out why the lever does not appear to turn despite the presence of equal and opposing forces.

Torque and Angular Momentum

Torque, sometimes called the moment of force, measures the tendency of a force to rotate an object about an axis or pivot. Torque is a vector quantity, and its magnitude is calculated as the product of the force and the perpendicular distance from the pivot point to the line of action of the force. Angular momentum, on the other hand, is a measure of the quantity of rotation of a body, which depends on the mass, shape, and velocity of the rotating object.

In our lever paradox, when analyzing the scenario from a distinct reference frame (frame S), one accounts for torque due to length contraction and high-speed effects. The combination of forces and the resulting torque would intuitively lead us to expect rotation – a couple caused by the torques. Yet, special relativity helps explain why, despite these torques, there is no angular rotation observed.

Momentum Density

Momentum density is a concept in physics that refers to the momentum per unit volume within a system. In the context of special relativity, momentum is not just a property of material objects but can also be associated with energy fields, including the electromagnetic field. This can lead to the idea of a 'non-material' momentum density, where energy can flow and carry momentum even in the absence of physical matter.

As related to the lever paradox, energy enters the lever at point C, which continually does work on the lever, while point B simultaneously does equal negative work. This flow of energy creates a non-material momentum density along the lever arm BC, toward B, in frame S. The continual flow and the associated momentum lead to an increase in angular momentum. This provides the resolution for why the lever accumulates angular momentum in frame S without physical rotation. It directly involves connecting the concept of momentum density to the rotational effects in a relativistic framework.