Circular Motion
1
Centripetal Acceleration
Imagine a marble sitting on a rotating turntable. The different vectors representing velocity for the travelling marble are shown below.
Notice that the size of the vector remains the same but the direction is constantly changing. Because the direction is changing, there is a ∆v and ∆v = vf
- vi , and since velocity is changing, circular motion must also be accelerated motion. vi
∆v vf -vi vf2
If the ∆t in-between initial velocity and final velocity is small, the direction of ∆v is nearly radial (i.e. directed along the radius). As ∆t approaches 0, ∆v becomes exactly radial, or centripetal. ∆v = vf - vi vi vf vf
∆v
-vi
Note that as ∆v becomes more centripetal, it also becomes more perpendicular with vf .
Also note that the acceleration of an object depends on its change in velocity ∆v;
i.e., if ∆v is centripetal, so is ‘a’.
From this, we can conclude the following for any object travelling in a circle at constant speed:
The velocity of the object is tangent to its circular path.
The acceleration of the object is centripetal to its circular path. This type of acceleration is called centripetal acceleration, or ac
.
The centripetal acceleration of the object is always perpendicular to its velocity at any point along its circular path. v ac ac v 3
To calculate the magnitude of the tangential velocity (i.e., the speed) of an object travelling in a circle:
• Start with d = vavt where ‘vav’ is a constant speed ‘v’
• In a circle, distance = circumference, so d = 2πr
• The time ‘t’ taken to travel once around the circular path is the object’s period ‘T’
• Therefore, the object’s speed is v =
Note that frequency f = 1
T
therefore v = 2πrf
There are two formulas for calculating the object’s centripetal acceleration:
• In terms of the object’s speed and radius of circular path: ac =
• In terms of the object’s period and radius of circular
Centripetal Acceleration
Imagine a marble sitting on a rotating turntable. The different vectors representing velocity for the travelling marble are shown below.
Notice that the size of the vector remains the same but the direction is constantly changing. Because the direction is changing, there is a ∆v and ∆v = vf
- vi , and since velocity is changing, circular motion must also be accelerated motion. vi
∆v vf -vi vf2
If the ∆t in-between initial velocity and final velocity is small, the direction of ∆v is nearly radial (i.e. directed along the radius). As ∆t approaches 0, ∆v becomes exactly radial, or centripetal. ∆v = vf - vi vi vf vf
∆v
-vi
Note that as ∆v becomes more centripetal, it also becomes more perpendicular with vf .
Also note that the acceleration of an object depends on its change in velocity ∆v;
i.e., if ∆v is centripetal, so is ‘a’.
From this, we can conclude the following for any object travelling in a circle at constant speed:
The velocity of the object is tangent to its circular path.
The acceleration of the object is centripetal to its circular path. This type of acceleration is called centripetal acceleration, or ac
.
The centripetal acceleration of the object is always perpendicular to its velocity at any point along its circular path. v ac ac v 3
To calculate the magnitude of the tangential velocity (i.e., the speed) of an object travelling in a circle:
• Start with d = vavt where ‘vav’ is a constant speed ‘v’
• In a circle, distance = circumference, so d = 2πr
• The time ‘t’ taken to travel once around the circular path is the object’s period ‘T’
• Therefore, the object’s speed is v =
Note that frequency f = 1
T
therefore v = 2πrf
There are two formulas for calculating the object’s centripetal acceleration:
• In terms of the object’s speed and radius of circular path: ac =
• In terms of the object’s period and radius of circular