Why is Tangential acceleration equal to alpha*Radius?
Tangential acceleration is equal to the product of angular acceleration and radius because of the relationship between linear and angular quantities in circular motion.
To understand why tangential acceleration is equal to alpha (angular acceleration) multiplied by radius, we need to consider a few concepts:
1. Circular motion: When an object moves in a circular path, it experiences acceleration, even if its speed remains constant. This acceleration is directed towards the center of the circle and is called centripetal acceleration.
2. Tangential acceleration: In addition to centripetal acceleration, the object also experiences acceleration tangential to its circular path. This tangential acceleration changes the object's linear velocity.
3. Angular acceleration: Angular acceleration, denoted by alpha (α), measures how quickly the angular velocity (rate of change of the angle) of an object changes over time. It is expressed in radians per second squared.
4. Radius: The radius (r) of a circular path is the distance between the center of the circle and the object moving along its circumference.
Now, let's see how tangential acceleration (at) relates to angular acceleration (alpha) and radius (r):
The tangential acceleration can be calculated as the rate of change of linear velocity, which is given by the equation:
at = dv/dt
But linear velocity (v) can be written in terms of angular velocity (ω) as:
v = ωr
Differentiating both sides with respect to time:
dv/dt = d(ωr)/dt
Here, d(ωr)/dt represents the rate of change of angular velocity multiplied by the radius. Recognizing that d(ωr)/dt is the definition of angular acceleration (alpha or α), we can substitute it to get:
at = alpha * r
Hence, tangential acceleration (at) is equal to the product of angular acceleration (alpha or α) and radius (r).
To summarize, tangential acceleration is equal to alpha (angular acceleration) multiplied by the radius because the change in linear velocity (tangential acceleration) in circular motion is determined by the rate of change of angular velocity (angular acceleration) and the distance from the center of the circle (radius).