topic 7: Motion in a Circle- Chapter 17

Question 1

what is angular displacement?

Answer 1

it is the angle θ through which the object has moved.

in a circle, the maximum angle can be 360 degrees or 2pi radians

Question 2

how to find angular displacement?

Answer 2

angle(in radians) = length of arc / radius

Question 3

how is 360 degrees = 2pi radians ?

Answer 3

a complete circle = 360 degrees
the length of arc of a complete circle = circumference of the circle = 2piradius

therefore,
angular displacement
of 360 degrees (in radians) = (2piradius) / radius
= 2pi rad

therefore,
360 degrees = 2pi rad

Question 4

define radian

Answer 4

one radian is the angle subtended at the centre of a circle by an arc of length equal to the radius of the circle 
that is,
2pi rad = 360 degrees 
1 rad = 360 / 2pi 
         = approx. 57.3 degrees

Question 5

the formula to convert degrees to radian

Answer 5

multiply by 2pi / 360 or pi/180

Question 6

the formula to convert radians to degrees

Answer 6

360 / 2pi or 180 / pi

Question 7

how to use newton’s laws of motion to explain circular motion?

Answer 7

consider the velocity of the object rather than its speed. this is important because velocity also takes in account the direction of the object.
therefore, the velocity of an object is constantly changing during circular motion.

Question 8

how is the direction of motion represented on a circular motion?

Answer 8

they are represented by arrows. these arrows are drawn as tangents to the circular motion.

Question 9

what is the difference between velocity and the angular velocity of an object moving in a circular motion?

Answer 9

the velocity of the circular motion is constantly changing due to the change in direction.

the angular velocity of an object is constant- because the object moves through the same angle every second.

Question 10

what is angular velocity?

Answer 10

the rate of change of angular position of a rotating body.

OR

how fast an object rotates or revolves relative to another point.

Question 11

how to find the angular velocity?

Answer 11

angular velocity (ω) = angular displacement / time 
                                                                            taken 

ω = rad s^-1

Question 12

what is the relationship between the speed of an object in circular motion, its angular velocity and the radius of the motion?

Answer 12

speed = angular velocity * radius

Question 13

how does Newton’s first law explain centripetal forces?

Answer 13

Newton’s first law states that an object remains at rest or in a stat of uniform motion unless it is object on by an external force.
since, an object moving at a steady speed in a circular motion is a body whose velocity is constantly changing; there is a resultant force acting on object moving in a circular motion.
this is the centripetal force.

Question 14

what is the centripetal force ?

Answer 14

the force directed towards the the centre of the circle acting on an object in circular motion is called centripetal force.

Question 15

what are causes the centripetal force?

Answer 15

the word “centripetal” is an adjective which describes that the force is making the object move in a circular path(directed towards the centre).

the force could actually be caused by gravitational, electrostatic, magnetic or any other reasons.

Question 16

how can we represent the change in velocity of an object moving along a circular path?

Answer 16

the change in velocity can be represented by a vector diagram.
the initial and final velocity are drawn tail-to-tail.

some properties of the diagram:

• the change vector is at right angles to the initial velocity.
• the change vector is directed towards the centre of the circle.

Question 17

what is the direction of acceleration of an object moving along a circular path?

Answer 17

a = change in velocity / time taken

therefore, it is in the same direction as velocity change.
that is, acceleration is directed towards the centre of the circle.

According to F = ma, acceleration of an object in circular motion will be in the same direction as the force ,i.e, centripetal force.

Question 18

what is the relationship between the direction of centripetal force, acceleration and the object’s velocity?

Answer 18

centripetal force and acceleration are in the same direction, i.e., towards the centre.

the centripetal force and acceleration are at right angles to the object’s velocity.

Question 19

how can a force change an object’s speed while also making it move in a circular path?

Answer 19

• the force must have a component the direction of the object’s initial velocity - to speed it up
• the force must also have a component which is 90 degrees to the object’s velocity- to make it move in circular motion

Question 20

what happens when a force exerted on a moving object is at 90 degrees to the object’s velocity?

Answer 20

the force acts to pull the object in a circular motion without ever changing its speed.
its component in the direction of the object’s velocity is
Fcos(90)= 0.

Question 21

how can we explain no change in speed of an object in a circular motion when a resultant force(centripetal force) is acting on it, in terms of work done?

Answer 21

work done = force * distance moved by the object in the direction of the force

the distance moved by the object in the direction of the force is zero; hence, work done is zero.

if no work is done, its kinetic energy must remain same; therefore, its speed remains unchanged.

Question 22

describe how each of the following quantities change as an object follows a circular path at a steady speed:

• speed
• velocity
• kinetic energy
• momentum
• centripetal force
• centripetal acceleration

Answer 22

• speed - constant
• velocity - constantly changing
• kinetic energy - constant
• momentum(mass*velocity) - constantly changing(magnitude remains constant but the direction is changing)
• centripetal force - same magnitude but different direction
• centripetal acceleration - same as centripetal force

Question 23

can we achieve a circular orbit under gravity at multiple speeds?

Answer 23

there is only one speed at which we can achieve a circular orbit under gravity.

Question 24

what is the relationship between the mass of an object, its speed in a circular motion and the centripetal force required?

Answer 24

the greater the mass, the greater the speed of the object in a circular motion.
therefore, the greater the centripetal force that is required to keep the object in a circular motion(F = ma)

Question 25

what is the relationship between the radius of the circle and the centripetal force required to keep the object in centripetal force?

Answer 25

if the radius of the circle is increased, the centripetal force required decreases.

Question 26

what are the different formulas to find the centripetal acceleration?

Answer 26

change in angle = change in velocity/ speed acceleration= change in velocity / time taken ACCELERATION = CHANGE IN VELOCITY/TIME TAKEN therefore, change in velocity = speed * change in angle acceleration = (speed * change in angle)/time taken A = (V*CHANGE IN ANGLE)/ T since, angular velocity = change in angle/ time taken acceleration = speed * angular velocity. A = V*ANGULAR VELOCITY since, speed = angular velocity* radius angular velocity = speed / radius therefore, acceleration = speed * (speed/ radius) = speed^2 / radius A = V^2 / R or acceleration = (angular velocity* radius) * angular velocity A = ANGULAR VELOCITY^2 * R

Question 27

what is the formula of centripetal force?

Answer 27

we can use Newton's second law of motion to find the formula of centripetal force. F = ma since, a=v^2 / r F = (m*v^2) / r or a = angular velocity^2 * r F = m*angular velocity^2 * r

Question 28

what is the direction of the centripetal force and the centripetal acceleration?

Answer 28

both centripetal force and centripetal acceleration are in the same direction , that is, toward the centre of the circle.

Question 29

how do you calculate the speed an object must have to orbit the earth under gravity?

Answer 29

centripetal force is the gravitational force on the object. therefore, mg = (m*v^2)/r g = v^2/r

Question 30

what is the speed an object must have to orbit around earth under gravity?

Answer 30

approximately 7.92 * 10^3 ms^-1

Question 31

what provides the centripetal force in the following scenarios: 1. a car cornering on a level road 2. a car cornering on a banked road 3. an aircraft banking 4. a stone being whirled in a horizontal circle on the end of the string 5. at the fairground

Answer 31

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Question 32

1. a car cornering on a level road

Answer 32

the normal force N balances the car's weight- therefore the car has no accleration in the vertical direction. the friction between the tyres and the road provides the centripetal force necessary for the car's circular motion. refer to fig 17.3 on page no.265

Question 33

2. a car cornering on a banked road

Answer 33

the normal contact force N has a horizontal component which can provide the centripetal force. the vertical component of N balances the weight of the car. ``` vertically = Ncosθ = mg horizontally = Nsinθ = (m*v^2)/r ```

Question 34

3. an aircraft banking

Answer 34

the vertical component of the lift force L on the wings balances its weight. the horizontal component of the lift L provides the centripetal force.

Question 35

4. a stone being whirled in a horizontal circle on the end of the string

Answer 35

the vertical component of the tension T is equal to the weight of the stone. the horizontal component of the tension provides the centripetal force.

Question 36

5. at the fairground

Answer 36

as the cylinder spins, the floor drops away. | the friction supports your weight and the normal force provides the centripetal force.

Question 37

what is the conical pendulum?

Answer 37

a weight is fixed on the end of a string or rod suspended from a pivot. Its construction is similar to an ordinary pendulum; however, instead of swinging back and forth, the bob of a conical pendulum moves at a constant speed in a circle with the string (or rod) tracing out a cone.

Question 38

what role does friction plays when a car moves in a circular motion on a banked road (bend) ?

Answer 38

if the car is travelling too slowly, it will start to slope downwards. the friction will act up the slope to keep the car on the course. if the car is travelling too fast, it will tend to slide up the slope. if friction is insufficient, it will move up the slope and come off the road.

Question 39

what happens when the speed of an object in a circular motion increases?

Answer 39

when the speed increases, a greater centripetal force is needed. if the centripetal force is not sufficient, the object starts to move in opposite direction of the centripetal force.