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Objectives
1.1 Uniform circular motion and centripetal force
a. Define the radian and express angular displacement in radians.
b. Understand and use the concept of angular velocity to solve problems.
𝜔 = ∆𝜃 /∆𝑡
c. Recall and use v = rω to solve problems.
d. Describe qualitatively motion in a curved path due to a perpendicular force, and understand the centripetal acceleration in the case of uniform motion in a circle.
e. Recall and use centripetal acceleration equations a = = r ù2
f. Recall and use centripetal force equations F = mrù2 = m
a. Radian Measure and angular displacement
If an object moves in a circle of radius r, then after travelling a distance s it has moved an angular displacement θ
A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius of the circle.
To obtain an angle in radians,(angular displacement) divide the length of the arc by the radius of the circle.
Angular displacement = θ =
Examples
a) Convert 65◦ to radians.
b) Convert 1.75 radians to degrees
b. Angular velocity
- Angular speed is the angle swept out by the radius of a circle per unit time.
- Angular velocity is the angular speed in a specified direction. It is the angle swept out by the radius of a circle per unit time in a specified direction
Angular velocity = =
For an object travelling at a constant speed v in a circle of radius r,
θ
,
S = and diving through by
= implying
that v = and
Note: In one complete revolution, the time taken is called period T, Thus angular speed =
Examples
1. A rocket makes a turn in a horizontal circle of radius 150 m. it is travelling at a speed of 240 m/s. Calculate the angular speed of the rocket.
= 1.6 rads-1
2. A car is travelling along a circular path with linear speed 18m/s and angular speed 0.30 rad/s. What is the radius of curvature of the track
From v = , r = = 18/0.30 = 60 m
3. A ball on a track travels round a complete loop in a time of 1.4s. Calculate the average angular speed of the ball.
angular speed = = 4.5 rads-1
c. Centripetal acceleration and centripetal force.
There are many everyday situations where objects travel in circular paths. The Earth spins on its axis and orbits the sun. A compact disc spins on disc players when the music is being played. Electrons orbit the nucleus of an atom. The hands of a mechanical clock follow a circular path as the time passes. Communicating satellites orbit the earth many times a day and winds within tropical cyclones move in circular paths. Car accidents often occur when the drivers of motor vehicles are trying to travel around a bend in a road at high speed. A vehicle that exceeds the safe speed limit can slip off the road and usually crash into anything along its path such as a tree or a fence. A vehicle travelling around a bend on a level road is moving along a circular path.
Uniform circular motion is the motion of a particle along a circular path with constant speed. It is accelerated motion; although speed is constant, velocity changes as direction changes
Consider a particle moving around a circle of radius r in the diagram below. At point A, the velocity of the particle is at a tangent to the circular path. When the particle reaches point B, its velocity still has the same magnitude, but its direction has changed. To find the acceleration, we need to determine the average change in velocity
The resultant’s direction is towards the centre of the circle, the acceleration and force experienced by the particle is towards the centre of the circle.
The acceleration due to a body in a circular motion is directed towards the centre of the circle and is called centripetal acceleration.
Centripetal acceleration a = v2 / r. = r ω2
The direction of the linear velocity is at a tangent to the circle described at that point. Hence it is sometimes referred to as the tangential velocity
Angular velocity ω is the same for every point in the rotating object, but the linear velocity v is greater for points further from the axis
A body moving in a circle at a constant speed changes velocity (since its direction changes). Thus, it always experiences an acceleration, a force and a change in momentum. The direction of resultant force (and hence acceleration) is directed towards the center
https://youtu.be/CzcmUiD39VI
EXAMPLES
1. A 3-kg rock swings in a circle of radius 5 m. If its constant speed is 8 m/s, what is the centripetal acceleration?
Centripetal acceleration a = v2 / r
r = 5 m v = 8 m/s
Centripetal acceleration = 8 2 / 5 = 12.8 m/s2
2. A car with mass 1000 kg drives through a curve with radius 200 m at speed 50 km/h. Find the centripetal acceleration
Centripetal acceleration a = v2 / r
r = 200 m v = 50 km/hr = 50/3.6 = 13.9 m/s
Centripetal acceleration = 13.92 / 200 = 0.96 m/s2
Centripetal force is the force acting on an object in circular motion. It acts along the radius of the circular path and towards the centre of the circle. It’s responsible for keeping the body moving along the circular path. It is the resultant of all forces that act on a system in circular motion. The centripetal force on a body is defined as the external force which causes the body to move in a circular path with a constant speed and acts along the radius and towards the centre of the circular path
Centripetal force F = m v2 / r. = m r ω2
As the centripetal force acts upon an object moving in a circle at constant speed, the force always acts inward as the velocity of the object is directed tangent to the circle. This would mean that the force is always directed perpendicular to the direction that the object is being displaced
EXAMPLES
1. A skater moves with 15 m/s in a circle of radius 30 m. The ice exerts a central force of 450 N. What is the mass of the skater?
V = 15 m/s r = 30 m F = 450 N
Centripetal force F = m v2 / r
450 = m x 152/ 30
M = 450/7.5 = 60 kg
2. A car with mass 1000 kg drives through a curve with radius 200 m at speed 50 km/h. Find the centripetal force
V = 50 m/s r = 200m mass = 1000 kg
Centripetal force F = m v2 / r
F = 1000 x 502/ 200
F = 12500 N
3. A wall exerts a 600 N force on an 80 kg person moving at 4 m/s on a circular platform. What is the radius of the circular path?
V = 4 m/s m = 80 Kg F = 600 N
Centripetal force F = m v2 / r
600 = 80 x 42/ r
r = 80 x 16 / 600
r = 2.3 m
WORKSHEET
1) What is the centripetal acceleration of the Moon towards the Earth? (Hint: you might need to look a few things up!)
2) Calculate the centripetal force acting on a 925 kg car as it rounds an unbanked curve with a radius of 75 m at a speed of 22 m/s.
3) A small plane makes a complete circle with a radius of 3282 m in 2.0 min. What is the centripetal acceleration of the plane?
4) A car with a mass of 833 kg rounds an unbanked curve in the road at a speed of 28.0 m/s. If the radius of the curve is 105 m, what is the average centripetal force exerted on the car?
5) An amusement park ride has a radius of 2.8 m. If the time of one revolution of a rider is 0.98 s, what is the speed of the rider?
6) An electron (m=9.11x10-31 kg) moves in a circle whose radius is 2.00 x 10-2 m. If the force acting on the electron is 4.60x10-14 N, what is its speed?
7) A 925 kg car rounds an unbanked curve at a speed of 25 m/s. If the radius of the curve is 72 m, what is the minimum coefficient of friction between the car and the road required so that the car does not skid?
8) A 2.7x103 kg satellite orbits the Earth at a distance of 1.8x107 m from the Earth’s centre at a speed of 4.7x103 m/s. What force does the Earth exert on the satellite?
9) A string can withstand a force of 135 N before breaking. A 2.0 kg mass is tied to the string and whirled in a horizontal circle with a radius of 1.10 m. What is the maximum speed that the mass can be whirled at before the string breaks?
10) A 932 kg car is traveling around an unbanked turn with a radius of 82 m. What is the maximum speed that this car can round this curve before skidding:
a) if the coefficient of friction is 0.95?
b) if the coefficient of friction is 0.40?
11) A motocross rider at the peak of his jump has a speed such that his centripetal acceleration is equal to g. As a result, he does not feel any supporting force from the seat of his bike, which is also accelerating at rate g. Therefore, he feels if there is ni force of gravity on him, a condition described as apparent weightlessness. If the radius of the approximately circular jump is 75.0 m, what is the speed of the bike?
1. 2.72 x 10-3 m/s2
2. (6.0x103 N)
3. (9.0 m/s2)
4. (6.2x103 N)
5. (18 m/s)
6. (3.18x107 m/s)
7. (0.89)
8. (3.3x103 N)
9. (8.62 m/s)
10. a. (28 m/s) b. (18 m/s)
11. 27 m/s