91Ó°ÊÓ

In uniform circular motion, centripetal acceleration is the key force that keeps a particle moving along a circular path. This force always acts towards the center of the circle, perpendicular to the particle's velocity, pulling the particle inward.
The formula for centripetal acceleration (\(a_c\)) is given by:

• \(a_c = \frac{v^2}{r}\), where \(v\) is the particle's velocity and \(r\) is the radius of the circle.

Understanding the direction and role of centripetal acceleration is crucial, as it dictates the radius and path of the motion. For instance, if a particle has a velocity of \(-5.00 \;\text{m/s}\) moving left and an upward acceleration of \(+12.5 \;\text{m/s}^2\), it implies that the center is located somewhere upward in relation to the particle's current position. This alignment helps to correctly determine the path and the direction of the centripetal force acting on the particle.

To figure out where the center of a circular path lies, particularly in a problem involving uniform circular motion, we examine both acceleration and velocity vectors. Given the coordinates for the center relate directly to these vectors:

• The velocity indicates the direction in which the particle moves tangent to the circle. In this exercise, the negative x-direction velocity suggests a path moving leftward.
• The acceleration vector acts perpendicular to the velocity, pointing towards the center. The positive y-direction of the acceleration shows that the center of our circle is somewhere above the particle's current location.

With the particle at \((4.00, 4.00)\) and not moving horizontally towards the center (due to the lack of x-acceleration), we know that the \(x\)-coordinate of the center remains the same as the particle's x-position. For the \(y\) position, using the radius derived from the centripetal acceleration, the center is found by adding this radius to the particle’s current y-coordinate.

In circular motion, velocity and acceleration work together, yet in contrasting directions. The velocity vector is always directed along the tangent of the circle at the particle's position. This means, at any point, if you imagine throwing an object from the particle’s position, it would travel along this tangent path.
On the other hand, acceleration in circular motion, specifically centripetal acceleration, acts inward towards the center of the circle. This might seem counterintuitive since we often think of acceleration as increasing speed. In circular motion, however, it changes the direction of the velocity rather than its magnitude.

• The velocity vector at the given instance was \(-5.00 \hat{i} \;\text{m/s}\), signifying leftward motion.
• The acceleration vector was \(+12.5 \hat{j} \;\text{m/s}^2\), directed upwards toward the center.

Together, these vectors form a dynamic duo in circular motion, ensuring the particle continues its circular path. Understanding how these directions and magnitudes interact helps in resolving where the center of the motion lies and how the path evolves over time.