CHAPTER 4 Motion in Two Dimensions

1. PROJECTILE MOTION 2. CALCULATION OF TIME OF FLIGHT 3. CALCULATION OF RANGE & MAX. RANGE. 4. CALCULTION OF ALTITUDE 5. CALCULATION OF TRAJECTORY 6. CIRCULAR MOTION7. ANFGULAR DISPLACEMENT8. ANGULAR VELOCITY 9. ANGULAR ACCELERATION 10. RELATION BETWEEN LINEAR AND ANGULAR VELOCITIES 11. CENTRIPETAL ACCELERATION12. CENTRIPETAL FOCE13. EQUATIONS 14. DIMENSION. 15. SHORT QUESTIONS AND ANSWERS

DESCRIPTIVE PARTProjectile motion:An object is thrown at certain angle with ground level; the motion of an object is under the influence of gravity. Such type of motion is called “Projectile motion”. The thrown object is called the projectile and its path is called the “trajectory”. Explanation: Let us consider a projectile, which is thrown at an angle with the horizontal, have initial take off velocity “vo”. A projectile carries out two motions:

1. A constant horizontal velocity vox = vo Cos , which remains constant, by neglecting air frictions,

2. and a vertical acceleration under the influence of gravity “g”. Such type of two dimensional motions is called “projectile motion”. Some examples of projectile motion are: A bullet firing from gun, Long jumping, Bomb dropping from airplane.Calculation of Time of flight:Total time taken by the projectile to cover the trajectory is called “Time of flight”, and denoted by “T”.

Where, at peak point and , just after leaving the ground.

Therefore,

Calculation of maximum range:The maximum horizontal distance covered by the projectile is called “Maximum Range”, denoted by “Rmax.”.If the value of Sin 2 is maximum, then range will be maximum.

It means,

Or , This is a time for up ward motion, to reach at the maximum height. And the

time required in down ward motion, to reach at the same level on ground is, .Hence, T = t + t

T = +

we get, , It is the total time of flight.Calculation of Range: Total horizontal distance covered by the projectile to hit the target, is called “Range”, denoted by “R”.

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The horizontal velocity component of projectile motion continues unchanged neglecting air friction. Hence the horizontal distance of flight is under constant horizontal velocity vox, given by,

, for uniform horizontal motion.

- - - - - - - - - ►eq. (a)

Or as total time of flight.

By putting the value of “T” in equation # (a), we get,

We get, , used for calculation of Range.

This shows that, if , then It is the “maximum range”.

Calculation of altitude:The maximum vertical distance covered by the projectile is called Altitude”.The down ward acceleration of the projectile is the same as free falling body and takes place independent of horizontal

motion. The Projectile has initial take off vertical velocity and final vertical velocity at peak point becomes zero Say ,vY = vf =0.

We know that,

- - - - - - - - - ►eq. (i)

Where,

Or

Hence,

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By putting the value of “t” in equation # (i), we get,

Calculation of Trajectory: A projectile covers a horizontal distance along with vertical is called “Trajectory”, means parabolic curve of projectile.

We know that,

Or - - - - - - - - - ►eq. (1)

By putting the value of “T” in equation # (1) we get,

, This equation is called “Parabola equation”.

Because, &

Uniform circular motion:1. Velocity remains constant in magnitude but varies in direction2. The acceleration is always normal to the velocity vector.3. The acceleration is always directed towards the centre of the circular path.Circular motion is caused when an object experiences just the right amount of net force directed toward the center of the circle or curve. The acceleration of an object in uniform circular motion is always toward the center of the circle.  The acceleration and the force causing it are called centripetal which means center-seeking. Centripetal force can be provided to a planet by gravity, to a car by friction between wheels and the road, or to a ball the the tension on a string. Since the direction of the velocity of the object is continuously changing toward the center of the circle, the object is said to experience centripetal acceleration, even though its speed is constant.

Some examples of circular motion:►A motion of the earth and other planets around sun, ► Motion of moon around earth, ►motion of electron around nucleus,

TECHNECHAL TERM RELATIVE DEFINITIONS

1. Rotational motion:A rigid object has circular motion about a fixed reference point pass through the rigid object. The motion of the rigid object as a whole is called “Rotational motion”.

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2. Angular motion: An object moves in a circle about fixed reference point .The angle made by the arc and the radius. Such a motion is called “Angular motion”.

3. Angular displacements:An angle subtended between an arc and radius of a circle is called “Angular displacement”, denoted by “”. This unit is called radian measure. It is dimensionless. The angular displacement of an object is the, ratio between arc length and radius of the circle when an object a moves from position A to position B in short time.

4. Conversion of degree into radians:The angle subtended by an arc of a circle whose arc length is equal to the radius of same circle is called “one radian”. The total circumference is (2r) is subtended by 360o

Hence, 360o = [2 ] radians

57. 3o = 1 radian1o = 0.01745 radiansAnd, one revolution = 2

5. Angular velocity: The average angular velocity of a rotating object is defined to be” the angular distance divided by the time taken to turn through this angle” denoted by “”, its unit is radians /sec.

Let us consider an object moving in a circle with uniform speed about fixed radius “r”. The angular displacement is from

position P to Q is in time “t”. The rate of change of angular displacement is the angular velocity. Its dimension is [ T-1].

Since,

It is vector quantity; direction is determined by right hand rule. If you curl the fingers of your right hand to follow the direction of the rotation, then the direction of the angular velocity vector is indicated by your right thumb

6. Instantaneous angular velocity: When time approaches to zero, the angular velocity at very short time interval is called “Instantaneous angular velocity”

7. Angular acceleration: The change of angular velocity in unit time is called “angular acceleration”. It is denoted by“”. Its unit is radians∕sec². Let us consider an object is moving in a circle, with angular velocity 1 in time t1, after t seconds the angular velocity becomes 2 in time t2. The change of angular velocity in unit time is “Angular acceleration”. Its dimension is T- ².

8. Instantaneous angular acceleration:When time approaches to zero, the angular acceleration at very short time interval is called “Instantaneous angular acceleration”.

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If the angular acceleration is uniform, we have that the average acceleration is the same as the instantaneous angular acceleration.The angular acceleration and angular velocity are vector quantities their direction is determined by “Right hand rule”.

According to this rule, “Place fingers of right hand on to wheel, by curling the fingers in the direction of rotation. The thumb point perpendicular to the wheel indicates the direction of angular quantities”. The clock-wise rotation is directed in to the paper, and anti- wise rotation is directed out of paper. The direction of angular velocity and angular acceleration is same in direction, angular velocity continuously increases in time, which is positive angular acceleration (+ ) and for the opposite directions the angular velocity continuously decreases in time, which is negative angular acceleration (-).

9. Relation between linear and angular velocities:The angular speed of the rotating object is to be related with the tangential speed of a position point A of an object Because, point A moves in a circle, the linear velocity “v: is always tangent to the circular path and hence is called tangential velocity. After “t” seconds the angular displacement is .

We know that,

By dividing “t” both sides we get,

r = v It is the magnitude of linear velocity with respect to angular velocity shows the relation between velocities

10. Tangential velocity:The product between radius of a circle and angular velocity is called “Tangential velocity”; denoted by “vt”.The tangential velocity of a point on a rotating rigid object equals the perpendicular distance of that point from the axis of rotation multiplied by the angular velocity. Therefore, although every point on the rigid object has the same angular velocity, not every point has the same linear speed because “r” is not the same for all points on the object. The direction is tangent to its circular path. Its magnitude is given by,

11. Tangential acceleration:The tangential acceleration of the point P can related to the angular acceleration of the rotating rigid object. The product between radius of a circle and angular acceleration is called “Tangential acceleration”; denoted by “at”. Suppose an object is moving about fixed axes, it changes its angular velocity in “t” seconds.

We know that

By dividing t both sides, We get,

Hence 12. Time period: The time required to complete one rotation is called “Time period”, denoted by “T”.Where s = v t 2 r = r T

Or DESCRIPTIVE PART

13. Centripetal acceleration:

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The acceleration perpendicular on to linear velocity, directed towards the centre, of circular path is called “Centripetal acceleration”.

Explanation:Let us consider a stone is attached at one end of a string and whirled round at constant speed in a circular path the direction of its velocity continuously changes. Therefore, it is said that a body moving on a circular path has acceleration.Mathematical derivation:

Suppose a stone is to be whirled in a circle of radius “R” at constant angular speed “”. Its linear tangential velocity “ ”

changes, in time t seconds. When, a stone change position from A to position B, than the velocities are and at the

respective position points. The magnitudes of velocities are same, = =

As the “O” is the origin of a circle, draw the representative lines “ ” for “ ” and “ ” for “ ”, according initial to

terminal rule, + = , is the resultant velocity. The change in velocity is between two given positions.

Hence, = –

The very small angular displacement between and is in very short time t.

Therefore, Divide both sides by t; it is time interval, in which velocity and angular displacement changed,

We get

The acceleration is towards the centre, known as centripetal acceleration.

Hence, This is magnitude of centripetal acceleration ac and always directed toward the centre of circle.

Because, v = r

Because,

Because,

The centripetal acceleration is also called “perpendicular acceleration” ( )

Thus,

And tangential acceleration” ” is the parallel acceleration “ ”. The magnitude and direction of resultant acceleration will be,

And 14. Centripetal force:

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A force pulling an object toward the center of a circular path as the object moves around the circle is called a centripetal force. An object can travel in a circle only.

Explanation:Let us consider a body of mass “m”, is attached at one end string which is whirled in a circle, the tension in the string provides a centripetal force, exerted by the string is unbalanced force. A body is moving along a circular path with uniform speed. The direction of velocity at any point on the circle is along the tangent to the circle. Thus the directions of motion of the body change from point to point. The centripetal force has no component acting on the body in the direction of motion. This means that the direction of centripetal force always perpendicular to the direction of motion of the body and directed towards centre of circle. By second law of motion, Fc = m ac Since, unbalanced force is not zero; the ball must have centripetal acceleration.

Hence,

Or

For example: Electron revolves round the nucleus. The force of attraction between the nucleus and the electron provides the required centripetal force. The planets move around the sun. The force of attraction between the sun and the planet provides the necessary centripetal force. The moon revolves round the earth. The force of attraction between the earth and the moon provides the required centripetal force. A stone at one end of string is whirled, a continuous centripetal force exerted inward by means of the string.

15. Centrifugal force: 16. The outward reaction acting on us is called the “Centrifugal force”, which tends to move the stone away from the circular path. Consider a stone whirled at the end of string. By means of the string we communicate the centripetal force to the stone. According, third law of motion the reaction of the stone acts outward direction. This shows that the necessary centripetal force to the stone through the string while an equal and opposite reaction of the stone acts on us directed away from the centre of circle ( F c = - F g ) . This reaction force is the “centrifugal force”. The centrifugal force acts as long as the centripetal force is present.

Equations of circular motion:

Equations

1. 2. 3. 4. 5.

6. 7. 8. 9. 10.

11. 12. 13. 14. 15.

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16. 7. 18. 19. 20.

21. 22.

DIMENSION

PHYSICAL QUANTITY SYMBOL DIMENSION UNIT

Range (R) meter

Time second

Altitude meter

Angular displacement = Dimensionless Radian or degree

Angular velocity per sec

Angular acceleration per sec-2

Tangential velocity m per sec

Tangential acceleration m per sec-2

Centripetal acceleration m per sec-2

Centripetal force Newton

Short questions and Answers

Q. #1: A bomber drops its bomb when, it is vertically above its target, it misses it. Explain the statement. A bomb followed trajectory, after dropping from the bomber. So that it can’t hit the target. So that a bomber drops its bomb when, it is vertically above its target, it misses it.

Q. #2: In broad jumping does it matter how you jump? What factors determine the span of the jump? In broad jumping it doesn’t matter how high we jump. The factors of span of the jump determine the initial. Velocity just after leaving the ground, angle of span of jump and acceleration due to gravity.

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Q. #3: Define the time flight and write its expression?

Total time of projectile to follow trajectory is called “time of flight”. Q. #4: Why a bomber does not drop the bombs when, it is vertically above the target?Because after dropping the bomb from the bomber a bomb followed trajectory. So that a bomber does not drop the bombs when, it is vertically above the target.

Q. #5: When a car takes a turn, the passenger’s experiences force acting on themselves away from of curve, why?Because, passengers’ in moving car try to balance themselves when a car takes a turn. So that, when a moving car turns round a corner to the left the centripetal force applied by the car and, the passenger’s in the opposite direction provides the centrifugal force.

Q. # 6: In what direction does the mud fly off the tires of moving car and why?In this case rotatary motion of the wheel provides the centrifugal force and the force between mud and wheel provides centripetal force. As the speed of wheel increases at a certain speed the reaction force exceeds the centripetal force .The mud will fly off the tangential direction to the wheel.

Q. #7: Does the horizontal velocity component of projectile motion remain constant, why?Yes, the horizontal velocity component of projectile motion remains constant, by neglecting air friction.

Q. # 8: A body is moving along a circle with constant speed i) is it moving with uniform velocity? ii) Is it moving with constant acceleration, what’s direction of acceleration?A body is moving along a circle with constant speed i) the direction of velocity changed at every point and the magnitude remains same. So that a body has no uniform velocity .ii) A body have constant acceleration that is centripetal. The direction is towards the centre of circle.

Q. # 9: Prove that, v = r

v = r or v = r

v = r .

we get, Hence shown

Q. #10: Explain the statement “work done by centripetal force is zero”.The centripetal force is always directed to the centre of a circle, which is perpendicular o to circumference. Work done is the product between force and displacement. Work = Fc d cos Work = Fc ( 2 r Cos 90o ) Work = 0, so that no work will be done due to centripetal force.

Q. #11: Angular velocity is a vector quantity. What is its direction?The direction of angular velocity is determined by right hand rule, according to this rule, place the fingers of right hand in the direction of rotation of disc .the thumb indicates the direction of angular velocity.

Q. #12: Show that angle of one radian is equal to 57.295o?

or 360 o = 360o = [2 ] radians

or = 1 radian

or = 1 radianHence, 57. 3o = 1 radian

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Q. #13: What is tangential velocity? The linear velocity at any point on the circumference of a circle, called “tangential velocity. It is produce between radius of a circle and angular velocity, say vt = r

Q. #14: Centripetal force and centrifugal reaction are equal in magnitude but opposite in direction. Do balance each other? The centripetal and centrifugal forces are equal in magnitude but opposite in direction along same line of action. According to the third law of motion, these forces are act as the action and reaction.

Q. #15: Is it harder for a car to turn round a corner at high speed than at low speed? Why. When a car turns round a corner the force of friction between the wheels of car and the road provides necessary centripetal force. As the speed increases the centripetal force increases and at a certain speed the centripetal force may become greater than the force of friction. Thus it is harder for a car to turn round a corner at a high speed.Q. #16: The experimental value of “g” would be greater at poles than at the equator, even if the earth will be a perfect sphere. Explain? The forces acting on a body situated at the equator are the force of gravity and centrifugal force. Hence the resultant force or

weight of a body F= m g- m r 2

Hence, the value of acceleration due to gravity at the equator, m g’= m g - m r 2

g’ = g - r 2 . “” is the angular velocity of the earth. Thus the value of “g” would be greater at poles than at the equator.

Q. #17: Why is the acceleration of a body moving in a circle directed towards the centre? Because, according to the second law of motion, the direction of force is same as the direction of acceleration. The centripetal acceleration is provided by centripetal force that is towards the centre on circle. Therefore, the acceleration of a body moving in a circle directed towards the centre. Fc = m ac

Q. #18: How would the weight of a man be affected if the earth stops rotating about its axis suddenly? The resultant force acting on the man situated at the equator towards the centre of the earth is given,

If the earth stops rotating suddenly the centrifugal force acting on him will become zero and its weight will be

.Thus weight of the man will increase.

Q. #19: Why are the near sights of a long-range rifle adjustable? There are two sights on the barrel of rifle. Near sight is close o eye of shooter and the far sight is at the end of the barrel. Since he range of bullet depends on the angle at which it is fired. The near sight can be adjusted to change the angle of firing. The firing form the rifle is the example of projectile motion. Thus long-range rifle the near sights can be adjusted according to the distance of the target.

Q. #20: How does air friction affects the path of a projectile. Air friction affects on the horizontal velocity component in projectile motion. This component may increases or decreases, depends on the direction of wind blowing

Q. # 21: The vertical velocity of projectile goes on increasing in the downward direction but its horizontal velocity remains constant. Why?Because, the vertical velocity component, of projectile is under the action of gravity which, goes on increasing in down ward motion direction but its horizontal velocity remains constant by neglecting air frictions.

Q. #22: Does the horizontal velocity of projectile remain exactly constant in air, also?If air and other force of frictions are not neglected then the horizontal velocity component of projectile motion exactly does not remain constant. Vox Vx

Q. #23: Show that a projectile can have same range for two angles of projections.

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The range for two angle of projections are same these are launch angle (say, 45o ) with vertical and take off angle with

horizontal. Q. #24: Define one radian.The circular length of a circle equal to the radius of circle is called “one radian”.

Q. #25 Give the relation between a) angular and linear velocities b) angular acceleration and linear acceleration c) angular velocity and time period.

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1. v = r , gives the relation between linear and angular velocities.2. a = r , gives the relation between linear and angular acceleration.

3. , gives the relation between time period and angular velocity. An object moving in a circle is accelerating; such objects are changing their velocity - either the speed (i.e., magnitude of the velocity vector) or the direction. An object undergoing uniform circular motion is moving with a constant speed. An object is accelerating due to its change in direction. The direction of the acceleration is inwards. The net force acting upon such an object is directed towards the center of the circle. The net force is said to be an inward or centripetal force. The net force directed perpendicular to the velocity; the object is always changing its direction and undergoing a centripetal acceleration.

1. Angle: the space between two lines or planes that intersect; the inclination of one line to another; measured in degrees2. Acute angle, one less than a right angle, or less than 90 degrees.3. Adjacent or Contiguous angles, such as have one leg common to both angles.4. Right angle, one formed by a right line falling on another perpendicularly, or an angle of 90 degrees.5. Solid angle, the figure formed by the meeting of three or more plane angles at one point.6. Spherical angle, one made by the meeting of two arcs of great circles, which mutually cut one another on the surface

of a globe or sphere.7. Internal angles, those which are within any right-lined figure.8. Mixtilineal angle, one formed by a right line with a curved line.9. Oblique angle, one acute or obtuse, in opposition to a right angle.10. Obtuse angle, one greater than a right angle, or more than 90 degrees.11. Angular distance: the angular separation between two objects as perceived by an observer; "he recorded angular

distances between the stars".12. Angular point, the point at which the sides of the angle meet; the vertex.13. Angular motion, the motion of a body about a fixed point or fixed axis, as of a planet or pendulum. It is equal to the

angle passed over at the point or axis by a line drawn to the body.14. Angular velocity, the ratio of angular motion to the time employed in describing.15. Optimum Angle: (1) The gliding angle at which the least height is lost in proportion to the ground covered. (2) The

angle of attack of an aerofoil at which the ratio of lift to drag is greatest. The angle is slightly less than 45 degrees. Example: Suppose the object is thrown 2 feet above the ground with an initial speed of 8 .05 ft/second. A 45 degree angle results in the object going 3.24 feet. The distance is 3.46 feet at the optimal angle of 30 degrees.

16. Angular velocity is defined as the time rate of change of angular displacement with respect to a point: Angular frequency ω (also referred to by the terms angular speed, radial frequency, circular frequency, and radian frequency) is a scalar measure of rotation rate. Angular frequency (or angular speed) is the magnitude of the vector quantity

17. Angular velocity. The term angular frequency vector is sometimes used as a synonym for the vector quantity angular velocity .

MCQ’s on Circular Motion1. Which one of the following statements about circular motion is correct?

A. The speed is changing all the time. B. The linear speed is constant but the acceleration is changing all the time. C. The linear speed is constant and the velocity is constant. D. The linear speed is constant but there is acceleration because the velocity is changing all the time, since there is a

constant change in direction.

2. Which one is correct about the centripetal force?A. It is inversely proportional to the velocity. B. It always acts at 90 degrees to the velocity, and away from the centre of the circle C. It always acts at 90 degrees to the velocity, and towards the centre of the circle D. It is balanced by the centrifugal force

3. A model aero plane is tethered to a post and held by a fine line. It flies in a horizontal circle. Then the line breaks. What direction will it fly in?

A. Directly to the centre of the circle. B. Directly away from the centre of the circle. C. In a circular path, as before D. In a straight line at a tangent.

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4. A satellite is in orbit. This is because:A. There is no gravity. B. The satellite is being held there by centrifugal force. C. Gravity is acting as a centripetal force. D. It is being attracted by the gravity of other objects in space.

5. Centripetal force can be worked out with the relationshipA. F = v^2/r B. F = mv^2/2 C. F = mv^2/r D. F = mr^2/vE.

6. Which one of the following statements is correct when a mass is made to move in a vertical circleA. The tension in the string is greatest at the bottom of the circle and least when at the top. B. The tension in the string is greatest at the top of the circle and least when at the bottom. C. The tension is provided by a combination of centrifugal force and weight. D. The weight acts in a direction away from the centre of the circle.

7. Angular velocity is measured in:A. Meters per second B. Degrees per second C. No units at all D. Radians per second

8. Angular velocity is used in preference to speed because:A. The speed depends on the radius. B. The centripetal force acts at 90 degrees to the speed. C. The speed does not change. D. Speed is harder to measure.

9. A car goes around a bend. What provides the centripetal force?A. Gravity B. Friction C. Electrostatic attraction D. No force at all.

10. A girl of mass 30 kg is on a roundabout of radius 5 m that makes one complete turn every 5 seconds. What is the centripetal force acting on her?

A. 38 N B. 47 N C. 236 N D. 300 N

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