Problem 36 - College Physics

Question

A ladybug starts at the center of a 12 -in.-diameter turntable and crawls in a straight radial line to the edge. While this is happening, the turntable turns through a \(45^{\circ}\) angle. (a) Draw a sketch showing the bug's path and the displacement vector for the bug's progress. (b) Find the magnitude and direction of the ladybug's displacement vector.

Step-by-Step Solution

Answer

The displacement magnitude is 6 inches, and the direction is 45°.

Step 1: Understand the Problem

The ladybug moves in a straight line from the center of the turntable to the edge. While it does this, the turntable also rotates by \(45^{\circ}\). This creates a path for the ladybug that is not just along the radius but also involves a rotational component.

Step 2: Draw the Path and Displacement

To visualize it, first draw a circle representing the turntable with a radius of \(6\) inches (since the diameter is \(12\) inches). Draw the initial straight line path from the center outward to the edge. Then draw the path as a straight line rotated \(45^{\circ}\) from the initial line. Finally, draw the displacement vector from the center to the point on the circle of radius \(6\) inches, \(45^{\circ}\) around from the original radial direction.

Step 3: Calculate Displacement Magnitude

The displacement magnitude is the distance from the center of the turntable to the position of the bug after both radial and rotational movement. Since the bug crawls directly to the edge, the magnitude of this displacement is simply the radius, which is \(6\) inches.

Step 4: Determine Direction of Displacement

The ladybug ends up \(45^{\circ}\) off from the original radial direction it started crawling along, due to the turntable's rotation. Therefore, the direction of the displacement vector is \(45^{\circ}\).

Key Concepts

Understanding Displacement Vectors

When we talk about displacement vectors, we're describing the shortest path from an initial position to a final position. In the context of the ladybug on a turntable, the displacement vector helps us understand how the bug moves within that scenario.
This vector is not just the straight line the bug crawls along; it combines both the linear and circular path the ladybug takes. In this problem, since the bug works its way from the center straight to the edge while the turntable rotates by \(45^{\circ}\), the displacement vector represents a combination of these changes.
The direction of the vector is crucial—here, it ends up at a \(45^{\circ}\) angle due to the rotation, yet the length of the vector—known as the magnitude—is simply the radius of the circle, in this case, \(6\) inches.

Circular Motion and Its Effects

Circular motion occurs when an object travels along a circular path. Even though the ladybug moves in a straight line, the involvement of a rotating turntable introduces a circular motion component.
This is an essential factor because it modifies the pathway that starts as a simple linear movement. As the turntable rotates, the ladybug's final position is not directly straight outward but influenced by this circular component.
To fully grasp this adjustment, consider the

• The initial radial path is in a straight line outwards.
• Including the turntable's rotation shapes this into a curvilinear trajectory.

Understanding this curvilinear path helps in determining the actual displacement vector, which is different from the naive linear prediction.

Geometry Unlocks Insights

Geometry plays a significant role in visualizing and solving physics problems like this one. It helps us break down complex movements into understandable components that can be easily calculated.
By framing the problem geometrically:

• First imagine a circle with a given diameter and radius – here, \(12\) inches and \(6\) inches respectively.
• Visualize the ladybug's radial path from center to edge.
• Account for the turntable's \(45^{\circ}\) rotation, showing a change in orientation.

This geometrical breakdown aids us in drawing paths and vectors, making it simpler to calculate the displacement magnitude and direction.

Vector Addition Simplified

Vector addition is a process used to determine a resulting vector from combining two or more vectors. In the ladybug scenario, while it initially might seem there's only one vector at play, combining radial and rotational movement is a form of vector addition.
Consider:

• The straight radial line is one vector pointing outward with a magnitude of \(6\) inches.
• The turntable's rotation adds another vector component, essentially shifting the entire system.

By understanding how these vectors add graphically, you can determine the precise direction and magnitude of the ladybug’s displacement vector. This combined vector respects both the movement direction and the turntable's influence on this path.