What is the angle made by the displacement vector of the point of a wheel (of radius R) initially in contact with the ground, with the x-axis, when the wheel rolls forward 3/4th of a revolution?

To determine the angle made by the displacement vector of a point on the circumference of a wheel with the x-axis after the wheel rolls forward three-quarters of a revolution, we can break down the problem step by step.

Understanding Wheel Movement

When a wheel rolls without slipping, every point on its circumference traces a circular path. The wheel's rotation can be visualized as a circular motion where the point of contact with the ground moves along the x-axis as the wheel rolls forward.

Visualizing the Rotation

Let's consider a wheel with a radius \( R \). Initially, when the wheel is at rest, the point on the circumference that is in contact with the ground is at the origin (0, 0) on a Cartesian plane. As the wheel rolls forward, this point lifts off the ground and moves in a circular arc.

Calculating the Angle

One complete revolution of the wheel corresponds to a rotation of \( 360^\circ \) or \( 2\pi \) radians. Therefore, three-quarters of a revolution is calculated as follows:

• Three-quarters of \( 360^\circ \) is \( 270^\circ \).
• In radians, this is \( \frac{3}{4} \times 2\pi = \frac{3\pi}{2} \) radians.

Position of the Point After Rolling

After rolling \( 270^\circ \) (or \( \frac{3\pi}{2} \) radians), the point that was initially at the bottom (0, 0) will have moved to a new position. To visualize this, think of the wheel as a circle centered at the origin. The point will now be located at:

• Coordinates: \( (0, -R) \)

Finding the Angle with the X-Axis

The displacement vector from the origin to the new position \( (0, -R) \) can be represented as a vector pointing straight down along the y-axis. The angle that this vector makes with the positive x-axis can be determined as follows:

• The vector points directly downward, which corresponds to an angle of \( 270^\circ \) or \( \frac{3\pi}{2} \) radians.

Final Answer

Thus, the angle made by the displacement vector of the point on the wheel with the x-axis after rolling forward three-quarters of a revolution is \( 270^\circ \) or \( \frac{3\pi}{2} \) radians. This angle indicates that the point is directly below the center of the wheel, aligned with the negative y-axis.