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Circular Motion
Circular motion is the motion of an object in a circular path or round a circle with constant speed but changing velocity due to change in the direction of travel of the object as the object moves round the circle.
When a stone is tied to one end of a rope and whirled along a horizontal circle so that it moves in a circular path (as shown in the figure below), then this stone is moving in a circular motion
As the stone is whirled, it sweep through angle θ and travel a circular distance S with constant speed. Its velocity changes because the stone changes its directions as it travels round the circle. As the stone moves round the circular path, and sweep through angle θ, the stone moves with angular velocity,w.
Relationship between linear speed and angular speed:
From the figure above, as the stone moves round the circular path and sweep through angle θ, distance S increases.
Therefore, we say that ,
Distance S is directly proportional to angle θ
Distance S & angle θ. therefore, S = r X θ.
Where
v is the linear velocity of the object that is moving in a circular path, measured in m/s.
r is the radius of the circular path which the object moved round, measured in meter.
ω is the angular velocity of the object, measured in radian per second ( rad/s )
- Let us make θ the subject, then we will get,
θ = s / r. ……… equation 1
Recall that, velocity = distance / time.
v = s / t ………. equation 2
Also, recall that, angular velocity = angle / time.
I.e ω = θ / t. ………… equation 3
Now, you substitute for θ of equation 1 in equation 3. Therefore you will get,
ω = θ / t - ω = θ X 1/t.
- ω = s/r X 1/t.
- ω X r = s/t … equation 4
Now , you substitute for v of equation 2 in equation 4. Therefore, you will get,
ω X r = s/t. - ω X r = v. Or. V = ω X r
Therefore, formula that connect v, r and ω is - V= ω X r
Example 1:
A stone of mass 0.5kg is tied to one end of a rope and whirled so that it moved in a circular path of radius 0.25m with a velocity of 15m/s for 2 minutes.
Calculate
- the angular velocity of the object
- angle sweep through,
- distance traveled.
Solution:
To calculate the angular velocity of the object
Step I: you extract the data given in the question:
Mass = 0.5kg, Velocity = 15m/s, radius of the circular path = 0.25m, time = 2 minutes.
time = 2 X 60 sec = 120 seconds
Step II: write down the formula for the calculation, which is
V= ω X r
Step III: substitute for the data in the equation. Therefore, you will get,
15m/s = ω X 0.25 .
Now you make ω the subject of the formula. Therefore, you will get,
ω = 15m/s / 0.25 m.
Answer : angular velocity ω = 60 radian per second
( 60 rad/s)
- b) Angle sweep through, I.e angle θ,
write down the formula for the calculation, which is
θ = s / r. ……… equation 1 and
ω = θ / t. ………… equation 2 above.
We can not use equation 1, we will use equation 2 because,the value of t is known and we have calculated ω from the calculation we did above. We can not use equation 1 because the value of s is not known or given in the question.
Therefore, ω = θ / t.
60 rad/s = ω = θ / 120 seconds
Make the subject, then we get,
θ = 60 rad/s X 120 seconds
θ = 2400 radians.
Now we have to change 2400 radians to degree.
360 degrees. = 2.π rad.
360° = 2 X 3.428
360° = 6.428 radians.
Now we will calculate how many degrees make 1 radian thus:
6.284 rad 360°,
1 rad = 360° / 6.284.
1 radian = 57.29°
Now we will convert 2400 radians to degrees by multiplying 2400 by 55.29°.
Therefore, 2400 rad = 2400 X 57.29° = 137496 °.
The angle θ which the object swept through = 137496°.
We can change this 137496° to numbers of revolutions by dividing it by 360° because 360° makes 1 revolution.
Therefore,
Number of revolutions = 137496 / 360 = 381.94 revolutions
c) Distance travelled.
We will use the formula, velocity = distance / time, to calculate distance travelled
( s=s/t)
Therefore, v = s / t.
15 m/s = distance / 120 seconds.
S = 15 m/s X 120seconds
Distance travelled = 180 meters