A particle is traveling in a circle of radius 2.5m and with an angular velocity of 10rad/s .if the particle begins to slow down with an angular acceleration of -1rad/s² ,after 55 seconds.

What is the centripetal acceleration and tangential acceleration of the particle.

To find the centripetal acceleration and tangential acceleration of a particle moving in a circular path, we need to understand a few key concepts related to circular motion. Let's break this down step by step.

Centripetal Acceleration

Centripetal acceleration is the acceleration that keeps an object moving in a circular path. It is directed towards the center of the circle. The formula for centripetal acceleration (a_c) is given by:

a_c = ω²r

Where:

• ω is the angular velocity (in radians per second)
• r is the radius of the circle (in meters)

In this case, the radius (r) is 2.5 m and the angular velocity (ω) is 10 rad/s. Plugging in these values:

a_c = (10 rad/s)² * 2.5 m

a_c = 100 rad²/s² * 2.5 m = 250 m/s²

Tangential Acceleration

Tangential acceleration (a_t) is the acceleration that occurs when the speed of the particle along the circular path changes. It can be calculated using the formula:

a_t = α * r

Where:

• α is the angular acceleration (in radians per second squared)
• r is the radius of the circle (in meters)

In this scenario, the angular acceleration (α) is -1 rad/s² (indicating that the particle is slowing down). Using the radius of 2.5 m:

a_t = (-1 rad/s²) * 2.5 m

a_t = -2.5 m/s²

Summary of Results

To summarize, after 55 seconds, the particle experiences:

• Centripetal Acceleration: 250 m/s²
• Tangential Acceleration: -2.5 m/s²

These values indicate that while the particle is maintaining a strong centripetal acceleration directed towards the center of the circle, it is also experiencing a negative tangential acceleration due to its deceleration. This means that the particle is slowing down as it moves along its circular path.