12.0.0 CIRCULAR MOTION:
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.
Demonstration of Circular Motion:
Tie a stone to one end of a rope and whirl it so that it moves in a circular path as shown below.
Explanation:
When a stone is tied to one end of a rope and whirl along a horizontal circle, the stone move in a circular path as sshown in the figure abelow. As the stone is whirled, it sweep through angle ® and travel a circular distance e 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 ®, 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®, distance S increases.
Therefore, we say that ,
Distance S is directly proportional to angle ®
Distance S & angle θ. therefore, S = r * θ.
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 * 1/t. ➡ ω = s/r * 1/t. ➡ ω * r = s/t ... equation 4
Now , you substitute for v of equation 2 in equation 4. Therefore, you will get,
ω * r = s/t. ➡ ω * r = v. Or. V = ω * r
Therefore, formula that connect v, r and ω is
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 )
APPLICATION OF FORMULA :
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:
1.
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 * 60 sec = 120 seconds
Step II: write down the formufor the calculation, which is
V= ω * r
Step III: substitute for the data in the equation. Therefore, you will get,
15m/s = ω * 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)
2. 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 * 120 seconds
θ = 2400 radians.
Now we have to change 2400 radians to degree.
360 degrees. = 2.π rad. ➡. 360° = 2 * 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 * 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
3. Distance travelled.
We will use the formula, velocity = distance / tim, to calculate distance travelled
( s=s/t)
Therefore, v = s / t. ➡ 15 m/s = distance / 120 seconds. ➡. S = 15 m/s * 1revolutions
Distance travelled = 180 meters
SUMMARY OF WORKING:
Data: Mass = 0.5kg, Velocity = 15m/s, radius of the circular path = 0.25m, time = 2min.
1. Formula: V= ω * r
Substitution: 15 m/s = ω * 0.25 m
Make the subject of the formula: ω = 15m/s / 0.25 m.
Answer: ω = 60 rad/s
2. Angle swept through
ω = θ / t. ➡. 60 rad/s = ω = θ / 120 seconds
Make the subject, then we get, ➡. θ = 60 rad/s * 120 seconds
θ = 2400 radians.
change 2400 radians to degree.
360 degrees. = 2.π rad. ➡. 360° = 2 * 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 * 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.
3. Distance travelled.
velocity = distance / time v = s / t.
Therefore,
v = s / t. ➡ 15 m/s = distance / 120 seconds. ➡. S = 15 m/s * 120 seconds
Distance travelled = 180 meters
CENTRIPETAL FORCE:
Centripetal force is an inward force that act on an object that is moving round a circle. The force help to stop the object from skidding out of the circular path.
s
FORMULA FOR CALCULATING CENTRIPETAL FORCE:
Centripetal force = m * v² /r ..... equation. 1
Where m is the mass of the object, v is the velocity of the object, r is the radius of the circular path
From the formula of force, force = mass * acceleration. ( f = m * a ). ..... equation. 2
I f we equate equation 1 and equation 2, then we will get,
M * a = M V² / r.
M will cancel M. Therefore, we will get, a = V² / r
Centripetal acceleration a = V² / r
WORKED EXAMPLE:
Calculate the magnitude if centripetal force on a car of mass 5.23kg that is moving round a circle of radius 0.5m with a a velocity of 10m/s.Also, determine the centripetal acceleration of the object.
SOLUTION:
Data: mass = 5.23kg, radius = 0.5m, velocity = 10 m/s
Formula : F = M * V² / r
Substitution:
F = 5.23kg * ( 10m/s )² / 0.5 ➡ F = 5.23 * 100 / 0.5 = 523 / 0.5 m = 100 / 0.5 = 1046N
Centripetal acceleration of the car:
Centripetal acceleration = V² / r =
C. A = ( 10m/s )² / 0.5 m = 100 / 0.5 = 200 m/s²
EXAMPLE 2:
If 200 N force act on a train of mass 34.5 kg that is moving round a circle with a velocity V, of radius 2.10 m, calculate the velocity of the train and hence what is its centripetal acceleration?
SOLUTION:
data: mass = 34.5 kg, velocity = V, radius = 2.10 m, force = 200N, c.a = ?
Formula: F = M * V² / r
Substitution: 200 = 34.5 kg * V² / 2.10.
Make V the subject: 200 * 2.10 / 34.5 = V² ➡ V² = 420 / 34.5 = 12.174
➡ V = √12.174. ➡ V = 3.49 m/s
Centripetal acceleration = V² / r ➡ c.a = 3.49² / 2.10 = 12.18 / 2.10 = 5.80 m/s²
C.A = 5.80 m/s²
CENTRIFUGAL FORCE:
Centrifugal force is an outward force which act on an object that moves round a circle. Centrifugal force is a balancing force which counterbalances the centripetal force so as to prevent the centripetal force from pushing the object to the centre of the circle. It is opposite to centripetal force.
FORMULA FOR CALCULATING CENTRIFUGAL FORCE:
Since centrifugal force is a balancing force, it is equal in magnitude but opposite in direction centripetal force.
Therefore,
Centripetal force = centrifugal force = M * V² / r
Same formulae is used to calculate centrifugal force
Centrifugal force = M * V² / r
ADVANTAGES OF CENTRIPETAL FORCE:
Centripetal force is responsible for the force that keeps the satellites in their orbits.
Centripetal force helps to maintain an object that is moving in a circular path
Centripetal force overcomes inertia
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.
Demonstration of Circular Motion:
Tie a stone to one end of a rope and whirl it so that it moves in a circular path as shown below.
Explanation:
When a stone is tied to one end of a rope and whirl along a horizontal circle, the stone move in a circular path as sshown in the figure abelow. As the stone is whirled, it sweep through angle ® and travel a circular distance e 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 ®, 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®, distance S increases.
Therefore, we say that ,
Distance S is directly proportional to angle ®
Distance S & angle θ. therefore, S = r * θ.
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 * 1/t. ➡ ω = s/r * 1/t. ➡ ω * r = s/t ... equation 4
Now , you substitute for v of equation 2 in equation 4. Therefore, you will get,
ω * r = s/t. ➡ ω * r = v. Or. V = ω * r
Therefore, formula that connect v, r and ω is
V= ω * rWhere
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 )
APPLICATION OF FORMULA :
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:
1.
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 * 60 sec = 120 seconds
Step II: write down the formufor the calculation, which is
V= ω * r
Step III: substitute for the data in the equation. Therefore, you will get,
15m/s = ω * 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)
2. 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 * 120 seconds
θ = 2400 radians.
Now we have to change 2400 radians to degree.
360 degrees. = 2.π rad. ➡. 360° = 2 * 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 * 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
3. Distance travelled.
We will use the formula, velocity = distance / tim, to calculate distance travelled
( s=s/t)
Therefore, v = s / t. ➡ 15 m/s = distance / 120 seconds. ➡. S = 15 m/s * 1revolutions
Distance travelled = 180 meters
SUMMARY OF WORKING:
Data: Mass = 0.5kg, Velocity = 15m/s, radius of the circular path = 0.25m, time = 2min.
1. Formula: V= ω * r
Substitution: 15 m/s = ω * 0.25 m
Make the subject of the formula: ω = 15m/s / 0.25 m.
Answer: ω = 60 rad/s
2. Angle swept through
ω = θ / t. ➡. 60 rad/s = ω = θ / 120 seconds
Make the subject, then we get, ➡. θ = 60 rad/s * 120 seconds
θ = 2400 radians.
change 2400 radians to degree.
360 degrees. = 2.π rad. ➡. 360° = 2 * 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 * 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.
3. Distance travelled.
velocity = distance / time v = s / t.
Therefore,
v = s / t. ➡ 15 m/s = distance / 120 seconds. ➡. S = 15 m/s * 120 seconds
Distance travelled = 180 meters
CENTRIPETAL FORCE:
Centripetal force is an inward force that act on an object that is moving round a circle. The force help to stop the object from skidding out of the circular path.
s
FORMULA FOR CALCULATING CENTRIPETAL FORCE:
Centripetal force = m * v² /r ..... equation. 1
Where m is the mass of the object, v is the velocity of the object, r is the radius of the circular path
From the formula of force, force = mass * acceleration. ( f = m * a ). ..... equation. 2
I f we equate equation 1 and equation 2, then we will get,
M * a = M V² / r.
M will cancel M. Therefore, we will get, a = V² / r
Centripetal acceleration a = V² / r
WORKED EXAMPLE:
Calculate the magnitude if centripetal force on a car of mass 5.23kg that is moving round a circle of radius 0.5m with a a velocity of 10m/s.Also, determine the centripetal acceleration of the object.
SOLUTION:
Data: mass = 5.23kg, radius = 0.5m, velocity = 10 m/s
Formula : F = M * V² / r
Substitution:
F = 5.23kg * ( 10m/s )² / 0.5 ➡ F = 5.23 * 100 / 0.5 = 523 / 0.5 m = 100 / 0.5 = 1046N
Centripetal acceleration of the car:
Centripetal acceleration = V² / r =
C. A = ( 10m/s )² / 0.5 m = 100 / 0.5 = 200 m/s²
EXAMPLE 2:
If 200 N force act on a train of mass 34.5 kg that is moving round a circle with a velocity V, of radius 2.10 m, calculate the velocity of the train and hence what is its centripetal acceleration?
SOLUTION:
data: mass = 34.5 kg, velocity = V, radius = 2.10 m, force = 200N, c.a = ?
Formula: F = M * V² / r
Substitution: 200 = 34.5 kg * V² / 2.10.
Make V the subject: 200 * 2.10 / 34.5 = V² ➡ V² = 420 / 34.5 = 12.174
➡ V = √12.174. ➡ V = 3.49 m/s
Centripetal acceleration = V² / r ➡ c.a = 3.49² / 2.10 = 12.18 / 2.10 = 5.80 m/s²
C.A = 5.80 m/s²
CENTRIFUGAL FORCE:
Centrifugal force is an outward force which act on an object that moves round a circle. Centrifugal force is a balancing force which counterbalances the centripetal force so as to prevent the centripetal force from pushing the object to the centre of the circle. It is opposite to centripetal force.
FORMULA FOR CALCULATING CENTRIFUGAL FORCE:
Since centrifugal force is a balancing force, it is equal in magnitude but opposite in direction centripetal force.
Therefore,
Centripetal force = centrifugal force = M * V² / r
Same formulae is used to calculate centrifugal force
Centrifugal force = M * V² / r
ADVANTAGES OF CENTRIPETAL FORCE:
Centripetal force is responsible for the force that keeps the satellites in their orbits.
Centripetal force helps to maintain an object that is moving in a circular path
Centripetal force overcomes inertia
According to Newton's Law of Inertia, an object in motion tends to follow a straight line.
If a force is applied to an object at an angle to the direction of motion, that force will overcome the object's inertia, such that it will follow a curved path, depending on the amount of the force and how long it is applied.
ADVANTAGES OF CENTRIFUGAL FORCE:
Centrifugal for e balances /centripetalforce.
Centrifugal is used in laboratory to separate solute from solution.
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