Problem 70 - University Physics with Modern Physics

Question

Three identical pucks on a horizontal air table have repelling magnets. They are held together and then released simultaneously. Each has the same speed at any instant. One puck moves due west. What is the direction of the velocity of each of the other two pucks?

Step-by-Step Solution

Answer

The other two pucks move 60° north of east and 60° south of east.

Step 1: Analyzing Initial Conditions

We are given three identical pucks that repel each other due to magnets. One puck moves due west after being released. Since they are identical and released simultaneously, the repulsion forces are equal in magnitude for each pair of pucks.

Step 2: Understanding Velocity Directions

The velocities of the pucks sum to zero because they start at rest and no external forces act on the system. Since one puck moves due west, the other two pucks must have velocities that result in a net zero vector sum along any axes to maintain balance.

Step 3: Determining Velocity Angle

To balance the westward velocity of the first puck, the other two pucks must move at equal angles from the east direction. We can use symmetry around the center of mass to determine the angles from the east direction.

Step 4: Calculating Angles

If the first puck's velocity is represented along the negative x-axis (west), the velocities of the other two pucks should lie in the positive half-plane. Each forms an angle of 120° with respect to the negative x-direction, or equivalently 60° north of east and 60° south of east.

Step 5: Checking Net Velocity

The resultant of the three velocities should sum to zero. The vector components of the other two pucks will cancel out the westward component when the angles are 60° north and 60° south of east, demonstrating that this configuration maintains equilibrium.

Key Concepts

Vector Sum

In physics, the concept of a vector sum is critical for understanding many phenomena. Vectors are quantities that have both magnitude and direction, such as force or velocity. When dealing with problems like the exercise at hand, it’s essential to know how to sum these vectors to analyze motion or force distribution.
To find the vector sum, you essentially "add" these vectors together following certain geometric rules, often visualized using arrows on a coordinate plane.

• Each vector is represented as an arrow pointing in a specific direction, with the length proportional to its magnitude.
• They can be added visually by placing the tail of one vector to the head of another.
• The resultant vector is drawn from the tail of the first vector to the head of the last vector, reflecting the cumulative effect.

In the context of the pucks on an air table, the vector sum of their velocities must equal zero, as they start from rest with no external forces acting on them. This symmetry ensures that the motion is perfectly balanced.

Velocity Direction

The direction of velocity is a crucial aspect in determining how an object moves. It’s not just about how fast something moves but also the path it takes. Velocity has two main components:
magnitude (speed) and direction. Understanding the direction component involves resolving which way an object will move in a plane or space.

• In polar coordinates, direction is often specified by an angle with respect to a reference direction (like east or north).
• The angle helps in predicting how the object will navigate over time from its point of release.
• Properly understanding velocity direction plays into predicting the final position of objects in motion.

In the exercise, one puck moves due west, indicating its direction directly. The other two must move in directions that cancel this westward movement precisely, resulting in net-zero velocity directionally, once again emphasizing balance through their equal angular placement from the east.

Magnetic Repulsion

Magnetic repulsion arises when like poles of magnets push away from each other. This principle is evident in the interacting pucks on the air table, where this repulsive force drives their motion.
The key factors influencing magnetic repulsion include:

• The strength of the magnets involved, which dictates how forcefully they repel each other.
• Distance between the magnets; the force generally diminishes with increased separation.
• The orientation of the magnets affects how the magnetic fields interact.

As the exercise demonstrates, the magnets in the pucks push them apart with equal force, ensuring each puck follows a path dictated by these repulsive dynamics. Understanding magnetic repulsion helps explain why the pucks move in specific and predictable directions once released.

Center of Mass Symmetry

Symmetry concerning the center of mass is a pivotal concept in many physics problems, including the one involving air table pucks. When objects are symmetric around their center of mass, it means:

• The system is balanced, and internal forces cancel each other out.
• The center of mass remains stationary if no external forces act.
• This symmetry is crucial for determining the paths of the objects affected, as seen with the puck exercise thoroughly.

In this example, the pucks possess symmetry at the point of release. Because of this symmetric distribution of mass and magnetic force, the pucks' directional velocities adjust naturally to maintain overall equilibrium. Such center of mass symmetry causes two pucks to move at complementary angles (60° north and south of east), counteracting the westward movement of the third puck, balancing the whole system.