Centripetal acceleration and circular motion

• GeneralOJB
In summary, increasing the radius in the centripetal acceleration formula |a| = ω^2*r also increases the acceleration, even if the angular speed remains constant. This can be explained by the fact that the linear velocity also increases with increasing radius, resulting in a larger distance traveled across the circle in the same amount of time. This intuitive understanding can also be seen in the alternative form of the formula, |a| = v^2 / r.
GeneralOJB
My question is about the centripetal acceleration formula |a| = ω^2*r. If we keep the angular speed constant then why does increasing the radius increase the centripetal acceleration? I don't find this intuitive because the velocity vector is being turned by the same amount each second, if ω is constant.

Also is there an intuitive way to think about how the units (m/s^2) relate to circular motion?

The linear velocity = ω * r. If r increases, then so does the linear velocity. Using the other form for centripetal acceleration:

|a| = ω^2 * r = v^2 / r

If r increases by a factor "c" and ω remains constant, then:

|a| = ω^2 * c * r = (c * v)^2 / (c * r) = c * v^2 / r

If ω is constant, the particle has to get from one side of the circle to the other in the same amount of time, ∏ω seconds.

The distance traveled "across the circle" is proportional to r, so it makes sense that the acceleration has to be bigger when r is bigger.

What is centripetal acceleration?

Centripetal acceleration is the acceleration that an object experiences while moving along a circular path. It is always directed towards the center of the circle.

How is centripetal acceleration calculated?

Centripetal acceleration can be calculated by taking the square of the velocity of the object and dividing it by the radius of the circle. This value is then multiplied by the constant pi (π).

What is the relationship between centripetal acceleration and speed?

The centripetal acceleration of an object is directly proportional to its speed. This means that as the speed of the object increases, so does the centripetal acceleration.

Why is centripetal acceleration necessary for circular motion?

Centripetal acceleration is necessary for circular motion because it is what keeps an object moving along a circular path. Without this acceleration, the object would continue to move in a straight line tangent to the circle.

What are some real-life examples of centripetal acceleration and circular motion?

Some examples of centripetal acceleration and circular motion include a car moving around a curve, a planet orbiting around a star, and a roller coaster going around a loop. Any motion that involves a circular path experiences centripetal acceleration.

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