MOTION IN A PLANE

In Fig. . (c), ∆ t Ž and the average acceleration becomes the instantaneous acceleration. It is directed towards the centre * .

Thus, we find that the acceleration of an object in uniform circular motion is always directed towards the centre of the circle. Let us now find the magnitude of the acceleration. The magnitude of a is, by definition, given by Let the angle between position vectors r and r ′ be ∆ θ . Since the velocity vectors v and v ′ are always perpendicular to the position vectors, the angle between them is also ∆ θ .

Therefore, the triangle CPP ′ formed by the position vectors and the triangle GHI formed by the velocity vectors v , v ′ and ∆ v are similar (Fig. .18a). Therefore, the ratio of the base-length to side-length for one of the triangles is equal to that of the other triangle. That is : = v Therefore, R If ∆ t is small, ∆θ will also be small and then arc PP ′ can be approximately taken to be| ∆ r |: r ≅ v t ≅ Therefore, the centripetal acceleration a c is : Fig.

. Velocity and acceleration of an object in uniform circular motion. The time interval ∆ t decreases from (a) to (c) where it is zero. The acceleration is directed, at each point of the path, towards the centre of the circle.

* In the limit ∆ t Ž , ∆ r becomes perpendicular to r . In this limit ∆ v → and is consequently also perpendicular to V . Therefore, the acceleration is directed towards the centre, at each point of the circular path. a c =    v = v /R ( .

) Thus, the acceleration of an object moving with speed v in a circle of radius R has a magnitude /R and is always directed towards the centre . This is why this acceleration is called centripetal acceleration (a term proposed by Newton). A thorough analysis of centripetal acceleration was first published in by the Dutch scientist Christiaan Huygens ( - ) but it was probably known to Newton also some years earlier. “Centripetal” comes from a Greek term which means ‘centre-seeking’.

Since v and R are constant, the magnitude of the centripetal acceleration is also constant. However, the direction changes — pointing always towards the centre. Therefore, a centripetal acceleration is not a constant vector. We have another way of describing the velocity and the acceleration of an object in uniform circular motion.

As the object moves from P to P ′ in time ∆ t (= t ′ – t ), the line CP (Fig. . ) turns through an angle ∆ θ as shown in the figure. ∆ θ is called angular distance.

We define the angular speed ω (Greek letter omega) as the time rate of change of angular displacement : ω ∆ t ( . ) Now, if the distance travelled by the object during the time ∆ t is ∆ s, i.e. PP ′ is ∆ s , then : s = ∆ but ∆ s = R ∆ θ . Therefore : ω v = R ω ( .

) We can express centripetal acceleration a c in terms of angular speed : c = ω ω c = ω ( . ) The time taken by an object to make one revolution is known as its time period T and the number of revolution made in one second is called its frequency ν (= / T ). However, during this time the distance moved by the object is s = π R . Therefore, v = π R / T = π R ν ( .

) In terms of frequency ν , we have ω = πν v = π R ν a c = π ν R ( . ) Example . An insect trapped in a circular groove of radius cm moves along the groove steadily and completes revolutions in s. (a) What is the angular speed, and the linear speed of the motion?

(b) Is the acceleration vector a constant vector ? What is its magnitude ? Answer This is an example of uniform circular motion. Here R = cm.

The angular speed ω is given by ω = π / T = π × / = . rad/s The linear speed v is : v = ω R = . s - × cm = . cm s - The direction of velocity v is along the tangent to the circle at every point.

The acceleration is directed towards the centre of the circle. Since this direction changes continuously, acceleration here is not a constant vector. However, the magnitude of acceleration is constant: a = ω R = ( . s – ) ( cm) = .

cm s - SUMMARY . Scalar quantities are quantities with magnitudes only. Examples are distance, speed, mass and temperature. .

Vector quantities are quantities with magnitude and direction both. Examples are displacement, velocity and acceleration. They obey special rules of vector algebra. .

A vector A multiplied by a real number λ is also a vector, whose magnitude is λ times the magnitude of the vector A and whose direction is the same or opposite depending upon whether λ is positive or negative. . Two vectors A and B may be added graphically using head-to-tail method or parallelogram method . .

Vector addition is commutative : A + B = B + A It also obeys the associative law : ( A + B ) + C = A + ( B + C ) . A null or zero vector is a vector with zero magnitude. Since the magnitude is zero, we don’t have to specify its direction. It has the properties : A + = A λ = A = .

The subtraction of vector B from A is defined as the sum of A and – B : A – B = A + ( – B ) . A vector A can be resolved into component along two given vectors a and b lying in the same plane : A = λ a + µ b where λ and µ are real numbers. . A unit vector associated with a vector A has magnitude and is along the vector A : ˆ n The unit vectors ɵ ɵ i, j, k are vectors of unit magnitude and point in the direction of the x-, y -, and z- axes, respectively in a right-handed coordinate system.

. A vector A can be expressed as i + = A where A x , A y are its components along x- , and y - axes. If vector A makes an angle θ with the x -axis, then A x = A cos θ , A y = A sin θ and , tan . .

Vectors can be conveniently added using analytical method . If sum of two vectors A and B , that lie in x-y plane, is R , then : ɵ , where, R x = A x + B x , and R y = A y + B y . The position vector of an object in x-y plane is given by r and the displacement from position r to position r’ is given by ∆ r = r ′ − r ′ − ′ − = ∆ + ∆ . If an object undergoes a displacement ∆ r in time ∆ t , its average velocity is given by v = .

The velocity of an object at time t is the limiting value of the average velocity as ∆ t tends to zero : v = ∆ t → . It can be written in unit vector notation as : ɵ where , , When position of an object is plotted on a coordinate system, v is always tangent to the curve representing the path of the object. . If the velocity of an object changes from v to v ′ in time ∆ t , then its average acceleration is given by: a v' The acceleration a at any time t is the limiting value of a as ∆ t Ž : In component form, we have : a where, a dv dt , a dv dt , a dv dt .

If an object is moving in a plane with constant acceleration and its position vector at time t = is r o , then at any other time t , it will be at a point given by: and its velocity is given by : v = v o + a t where v o is the velocity at time t = In component form : ox v t oy ox oy Motion in a plane can be treated as superposition of two separate simultaneous one- dimensional motions along two perpendicular directions . An object that is in flight after being projected is called a projectile . If an object is projected with initial velocity v o making an angle θ o with x- axis and if we assume its initial position to coincide with the origin of the coordinate system, then the position and velocity of the projectile at time t are given by : x = ( v o cos θ o ) t y = ( v o sin θ o ) t − ( / ) g t v x = v ox = v o cos θ o v y = v o sin θ o − g t The path of a projectile is parabolic and is given by : cos gx x – v The maximum height that a projectile attains is : h 2g = ( q The time taken to reach this height is : The horizontal distance travelled by a projectile from its initial position to the position it passes y = during its fall is called the range , R of the projectile. It is : sin2 .

When an object follows a circular path at constant speed, the motion of the object is called uniform circular motion . The magnitude of its acceleration is a c = v /R . The direction of a c is always towards the centre of the circle. The angular speed ω , is the rate of change of angular distance.

It is related to velocity v by v = ω R . The acceleration is a c = ω R . If T is the time period of revolution of the object in circular motion and ν is its frequency, we have ω = π ν, v = πν R, a c = π ν R POINTS TO PONDER . The path length traversed by an object between two points is, in general, not the same as the magnitude of displacement.

The displacement depends only on the end points; the path length (as the name implies) depends on the actual path. The two quantities are equal only if the object does not change its direction during the course of motion. In all other cases, the path length is greater than the magnitude of displacement. .

In view of point above, the average speed of an object is greater than or equal to the magnitude of the average velocity over a given time interval. The two are equal only if the path length is equal to the magnitude of displacement. . The vector equations ( .33a) and ( .34a) do not involve any choice of axes.

Of course, you can always resolve them along any two independent axes. . The kinematic equations for uniform acceleration do not apply to the case of uniform circular motion since in this case the magnitude of acceleration is constant but its direction is changing. .

An object subjected to two velocities v and v has a resultant velocity v = v + v . Take care to distinguish it from velocity of object relative to velocity of object : v = v − v . Here v and v are velocities with reference to some common reference frame. .

The resultant acceleration of an object in circular motion is towards the centre only if the speed is constant. . The shape of the trajectory of the motion of an object is not determined by the acceleration alone but also depends on the initial conditions of motion ( initial position and initial velocity). For example, the trajectory of an object moving under the same acceleration due to gravity can be a straight line or a parabola depending on the initial conditions.

EXERCISES . State, for each of the following physical quantities, if it is a scalar or a vector : volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity. . Pick out the two scalar quantities in the following list : force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, relative velocity.

. Pick out the only vector quantity in the following list : Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge. . State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (a) adding any two scalars, (b) adding a scalar to a vector of the same dimensions , (c) multiplying any vector by any scalar, (d) multiplying any two scalars, (e) adding any two vectors, (f) adding a component of a vector to the same vector.

. Read each statement below carefully and state with reasons, if it is true or false : (a) The magnitude of a vector is always a scalar, (b) each component of a vector is always a scalar, (c) the total path length is always equal to the magnitude of the displacement vector of a particle. (d) the average speed of a particle (defined as total path length divided by the time taken to cover the path) is either greater or equal to the magnitude of average velocity of the particle over the same interval of time, (e) Three vectors not lying in a plane can never add up to give a null vector. .

Establish the following vector inequalities geometrically or otherwise : (a) | a + b | < | a | + | b | (b) | a + b | > || a | −−−−− | b || (c) | a −−−−− b | < | a | + | b | (d) | a −−−−− b | > || a | −−−−− | b || When does the equality sign above apply? . Given a + b + c + d = , which of the following statements are correct : (a) a , b , c , and d must each be a null vector, (b) The magnitude of ( a + c ) equals the magnitude of ( b + d ), (c) The magnitude of a can never be greater than the sum of the magnitudes of b , c , and d , (d) b + c must lie in the plane of a and d if a and d are not collinear, and in the line of a and d , if they are collinear ? .

Three girls skating on a circular ice ground of radius m start from a point P on the edge of the ground and reach a point Q diametrically opposite to P following different paths as shown in Fig. . . What is the magnitude of the displacement vector for each ?

For which girl is this equal to the actual length of path skate ? . A cyclist starts from the centre O of a circular park of radius km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. .

. If the round trip takes min, what is the (a) net displacement, (b) average velocity, and (c) average speed of the cyclist ? Fig. .

. On an open ground, a motorist follows a track that turns to his left by an angle of after every m. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.

. A passenger arriving in a new town wishes to go from the station to a hotel located km away on a straight road from the station. A dishonest cabman takes him along a circuitous path km long and reaches the hotel in min. What is (a) the average speed of the taxi, (b) the magnitude of average velocity ?

Are the two equal ? . The ceiling of a long hall is m high. What is the maximum horizontal distance that a ball thrown with a speed of m s - can go without hitting the ceiling of the hall ?

. A cricketer can throw a ball to a maximum horizontal distance of m. How much high above the ground can the cricketer throw the same ball ? Q Fig.

. . A stone tied to the end of a string cm long is whirled in a horizontal circle with a constant speed. If the stone makes revolutions in s, what is the magnitude and direction of acceleration of the stone ?

. An aircraft executes a horizontal loop of radius . km with a steady speed of km/h. Compare its centripetal acceleration with the acceleration due to gravity.

. Read each statement below carefully and state, with reasons, if it is true or false : (a) The net acceleration of a particle in circular motion is always along the radius of the circle towards the centre (b) The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point (c) The acceleration vector of a particle in uniform circular motion averaged over one cycle is a null vector . The position of a particle is given by . .

. where t is in seconds and the coefficients have the proper units for r to be in metres. (a) Find the v and a of the particle? (b) What is the magnitude and direction of velocity of the particle at t = .

s ? . A particle starts from the origin at t = s with a velocity of . j ɵ m/s and moves in the x-y plane with a constant acceleration of ( .

. m s - . (a) At what time is the x - coordinate of the particle m? What is the y -coordinate of the particle at that time?

(b) What is the speed of the particle at the time ? . ɵ i and ɵ j are unit vectors along x - and y - axis respectively. What is the magnitude and direction of the vectors ɵ + , and ɵ − ?

What are the components of a vector A = ɵ + along the directions of ɵ + and ɵ − ? [You may use graphical method] . For any arbitrary motion in space, which of the following relations are true : (a) v average = ( / ) ( v ( t ) + v ( t )) (b) v average = [ r ( t ) - r ( t ) ] /( t – t ) (c) v ( t ) = v ( ) + a t (d) r ( t ) = r ( ) + v ( ) t + ( / ) a t (e) a average =[ v ( t ) - v ( t )] /( t – t ) (The ‘average’ stands for average of the quantity over the time interval t to t ) . Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (a) is conserved in a process (b) can never take negative values (c) must be dimensionless (d) does not vary from one point to another in space (e) has the same value for observers with different orientations of axes.

. An aircraft is flying at a height of m above the ground. If the angle subtended at a ground observation point by the aircraft positions .