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Now we shall discuss another example of two-dimensional motion that is motion of a particle on a circular path. This type of motion is called circular motion.

Consider a particle P is moving on circle of radius r on X – Y plane with origin O as centre.

The position of the particle at a given instant may be described by angle q, called angular position of the particle, measured in radian. As the particle moves on the path, its angular position (q) changes. The rate of change of angular position is called angular velocity (w) measured in radian per second.

The rate of change of angular velocity is called angular acceleration, measured in rad/s². Thus, the angular acceleration is

RELATION BETWEEN DIFFERENT PARAMETERS OF CIRCULAR MOTION

ü        It is easy to derive the equations of rotational kinematics for the case of constant angular acceleration with fixed axis of rotation. These equations are of the same form as those for one-dimensional translational motion.

w = w₀ +αt                                                             …(i)

Φ = Φ₀ + w₀t + αt²/2                                 …(ii)

w² = w₀² + 2α (Φ- Φ₀)                             …(iii)

Φ= Φ₀ + (w₀ +w)/(2t)                               …(iv)

Here,Φ is the initial angle and w₀ is the initial angular speed.

Illustrations

1.     (a) What is the angular velocity of the minute and hour hands of a clock?

(b) Suppose the clock starts malfunctioning at 7 AM which decelerates the minute hand at the rate of 4p radians/day. How much time would the clock loose by 7 AM next day?

Sol. Angular speed of

minute hand : wmh = 2Π rad/hr

= 48Π rad/day = (Π/1800) rad/sec

hour hand : whh = (Π/6) rad/hr

= 4Π rad/day = (Π/21600) rad/sec

(b)     Assume at t = 0, Φ₀ = 0, when the clock begins to malfunction. Use equation (ii) to get the angle covered by the minute hand in one day.

Hence the minute hand complete 23 revolutions, so the clock losses 1 hour.

along the vertical y-axis with a uniform downward acceleration ‘g’ and

• along the horizontal x-axis with a uniform velocity forward.

Consider a particle projected with an initial velocity u at an angle α with the horizontal x-axis as shown in figure 2.17. Velocity and accelerations can be resolved into two components:

Velocity along x-axis = u_x = u cos α

Acceleration along x-axis a_x = 0

Velocity along y-axis = u_y = u sin α

Acceleration along y-axis a_y = -g

Here we use different equation of motions of one dimension derived earlier to get the different parameters.

v² = v₀² – 2g (y – y₀)                                …(C)

Total Time of flight

When body returns to the same horizontal level, the resultant displacement in vertical y-direction is zero. Use equation B.

Therefore,             0 = (u sin α) t – (½)gt²

or                   t = 2uSinα / g

(as t cannot equal to 0)

Horizontal Range

Horizontal Range (OA)  = Horizontal velocity × Time of flight

=        u cos α × 2u sin α/g

= u² sin 2α/g

=

Maximum Height

At the highest point of the trajectory, vertical component of velocity is zero.

Therefore              0 = (u sin α)² – 2g H_max

or,           H_max = u² sin² α/ 2g

Equation of trajectory

Assuming the point of projection as the origin of co-ordinates and horizontal direction as the x-axis and vertical direction as the y-axis. Let P (x, y) be the position of the particle at instant after t second.

Then x = u cos α.t                and         y = u sin α.t – 1/2 gt²

Eliminating  ‘t’ form the above equations, we get,

y = x tan α – gx²/ 2u² cos² α

This is the equation of trajectory which is a parabola  (y = ax + bx²).

Illustrations

1.      A gun moving at a speed of 30m/sec fires at an angle 300 with a velocity 150m/s relative to the gun. Find the distance between the gun and the projectile when projectile hits the ground . (g = 10 m/sec)

Sol. Vertical component of velocity = 150 sin 300 = 75 m/sec

Horizontal component of velocity relative to gun = 150 cos 300

=                       =   75 √3 m/sec

Horizontal component of velocity relative to ground

=                              = 75 √3 + 30 ≈ 160 m/sec

Time of flight = (2 * 75 )/g = 15 sec

Range of projectile              = 160 × 15 = 2400 m

Distance moved by the gun and projectile = 2400 – 450 = 1950 m.

Consider a particle projected horizontally with a velocity u  from a point O as shown in figure 2.18.

Assuming the point of projection O as the origin of coordinates and horizontal direction as the X-axis and vertical direction as Y-axis. Let P (x, y) be the position of the particle after t seconds.

x = horizontal distance covered in time t = ut. …(1)

y = vertical distance covered in time t  = ½gt² ….(2)

Eliminate t from equations  (1) and (2) then

We get, y = (1/2 ) (g/u²) x²

This is the equation of parabola passing through the origin, with its vertex at the origin O. Hence the trajectory is a parabola.

• The position of object can change on a straight line (like on x-axis with respect to origin) or on a plane with respect to some fixed point or frame. So we can define motion as follows:
• An object or a body is said to be in motion if its position continuously changes with time with reference to a fixed point (or fixed frame of reference)
• When the position of object changes on a straight line i.e. motion of object along straight line is called motion in one dimension.
• To understand the essential concepts of one dimensional motion we have to go through some basic definitions.
• One can see the platform from a running train, and it seems that all the objects placed on platform are continuously changing their position. But one, who is on platform, concludes that the objects on the platform are at rest. It means if we will take the train as reference frame the objects are not stationary and taking reference frame as platform the objects are stationary. So the study of motion is a combined property of the object under study and the observer. Hence there is a need to define a frame of reference under which we have to study the motion of an object. We can define the frame of reference as follows:
• A frame of reference is a set of coordinate axes which is fixed with respect to a space point (a body or an object can also be treated as a point mass and therefore it can become a site for fixing a reference frame), which we have arbitrarily chosen as per our observer’s requirement. The essential requirement for a frame of reference is that, it should be rigid.
• The position of an object is defined with respect to some frame of reference. As a convention, we define position of a point (essentially we treat body as a point mass) with the help of three co-ordinates X, Y and Z. Hence (X, Y, Z) is a set of coordinate axes representing a 3-dimensional space and each point in this space can be uniquely defined with the help of a set of X, Y and Z coordinate, all three axes being mutually perpendicular to each other. The line drawn from origin to the point represents the position vector of that point.

In the figure 2.1, the position of a point P is specified and

is called the position vector.

• Consider a case in which the position of an object changes with time. Suppose at certain instant ‘t’ the position of an object is x₁ along the x axis and some other instant ‘T’ the position is x ₂ then the displacement Δx is defined as

Δx = x₂ – x₁

• It can be seen in the figure 2.2 where x₁ and x₂ are instantaneous position of the object at time t and T respectively.
• Now consider the motion of a point A with respect to a reference point O. The motion of point A makes its radius vector vary in the general case both in magnitude and in direction as shown in figure 2.3. Suppose the point A travels from point 1 to point 2 in the time interval Δt.

Distance and Displacement

• To understand the difference between distance and displacement, we study the motion of vertical throw of a ball with respect to point O to height h.
• After some time it will come again to the same point O. The displacement of ball is zero but there is some distance traversed by the ball. It’s because distance is a scalar quantity but displacement is a vector quantity.

Speed and Velocity

• Speed is the rate of change of distance without regard to directions. Velocity is the rate at which the position vector of a particle changes with time. Velocity is a vector quantity whereas speed is scalar quantity but both are measured in the same unit m/sec.

• Imagine flying birds, moving planets, gushing air, flowing water and so on, all these phenomena are happening around us continuously. All the above mentioned phenomena can be summarized in one word “motion”.
• When something moves, there are several factors, which can be observed. If it moves, it will cover some distance, the distance covered may be different in different time periods. It may be moving slow or fast. Further its speed or velocity may be changing with time. All these factors depend on each other and we need to study their relationships.             Various kinds of motion can be systematically grouped under few broad categories. The point of distinction is made on grounds of the velocity vector. On these grounds motion can be:
• one dimensional, where only one dimension is required to describe the motion
• two-dimensional, where we require two dimensions to describe the motion of the particle
• three-dimensional, where three dimensions are required.

Pre-requisite

State of Rest and Motion

• If the position of a particle is changing with respect to its surroundings in a given time interval, we say the particle is in motion and on the other hand if the position particle is not changing with respect to its surroundings, we can say the particle is at rest.

Position

The position of a particle refers to its location in the space at a certain moment. A vector joining the moving particle with the origin can describe the position of that particle at a particular moment.

Displacement

It is the vector joining the initial position of the particle to its final position in a given time interval.

Velocity

The rate of change of position of a moving particle is known as its velocity.

Acceleration

The rate of change of velocity with time is called the acceleration.

Distance and Displacement

• To understand the difference between distance and displacement, we study the motion of vertical throw of a ball with respect to point O to height h.
• After some time it will come again to the same point O. The displacement of ball is zero but there is some distance traversed by the ball. It’s because distance is a scalar quantity but displacement is a vector quantity.

Speed and Velocity

• Speed is the rate of change of distance without regard to directions. Velocity is the rate at which the position vector of a particle changes with time. Velocity is a vector quantity whereas speed is scalar quantity but both are measured in the same unit m/sec.