Kinematics

Dr. Vishesh Ranjan Kar Assistant Professor Department of Mechanical Engineering NIT Jamshedpur

Dynamics of Particles

Introduction

Example

Determine: • velocity V and elevation y above ground at time t, • highest elevation reached by ball and corresponding

value of time. • time when ball will hit the ground and corresponding

velocity. Draw the V-t and y-t curves.

A ball is tossed with velocity 10 m/s directed vertically upward from a window located at 20 m above ground. Knowing that the acceleration of the ball is constant and equal to 9.81 m/s2 downward.

The motion of the slider A is defined by the relation x = 500sin kt,

where x and t are expressed in millimeters and seconds, respectively,

and k is a constant. Knowing that k = 10 rad/s, determine the

position, the velocity, and the acceleration of slider A when t = 0.05 s.

Automobiles A and B are traveling in adjacent highway lanes and at t = 0 have the positions and speeds shown. Knowing that automobile A has a constant acceleration of 0.6 m/s2 and that B has a constant deceleration of 0.4 m/s2, determine (a) when and where A will overtake B, (b) the speed of each automobile at that time.

Two automobiles A and B are approaching each other in adjacent

highway lanes. At t = 0, A and B are 1 km apart, their speeds are

vA = 109 km/hr and vB = 62.5 km/hr, and they are at points P and Q,

respectively. Knowing that A passes point Q 40 s after B was there

and that B passes point P 42 s after A was there, determine (a) the

uniform accelerations of A and B, (b) when the vehicles pass each

other, (c) the speed of B at that time.

Block C moves downward with a constant velocity of 2 m/s.

Determine (a) the velocity of block A, (b) the velocity of block D.

Curvilinear Motion A particle moving along a curve other than a straight line is said to be in curvilinear motion.

Velocity of a particle is a vector tangent to the path of the particle

Acceleration is not tangent to the path of the particle

A projectile is fired from the edge of a 150-m cliff with an initial velocity of 180 m/s at an

angle of 30° with the horizontal. Neglecting air resistance, find (a) the horizontal distance

from the gun to the point where the projectile strikes the ground, (b) the greatest elevation

above the ground reached by the projectile

A projectile is fired with an initial velocity of 800 m/s at a target B located 2000m above the gun A and at a horizontal distance of 12,000 m. Neglecting air resistance, determine the value of the firing angle α.

Tangential component of the acceleration is equal to the rate of change of the speed of the particle. Normal component is equal to the square of the speed divided by the radius of curvature of the path at P.

A motorist is traveling on a curved section of highway of radius 2500 ft at the speed of

60 mi/h. The motorist suddenly applies the brakes, causing the automobile to slow

down at a constant rate. Knowing that after 8 s the speed has been reduced to 45 mi/h,

determine the acceleration of the automobile immediately after the brakes have been

applied.

The position of the particle P is defined by polar coordinates r and θ. It is then convenient to resolve the velocity and acceleration of the particle into components parallel and perpendicular to the line OP.

Unit vector er defines the radial direction, i.e., the direction in which P would move if r were increased and θ were kept constant.

The unit vector eθ defines the transverse direction, i.e., the direction in which P would move if θ were increased and r were kept constant.

Where -er denotes a unit vector of sense opposite to that of er

Using the chain rule of differentiation,

Using dots to indicate differentiation with respect to t

To obtain the velocity v of the particle P, express the position vector r of P as the product of the scalar r and the unit vector er and differentiate with respect to t:

Differentiating again with respect to t to obtain the acceleration,

The scalar components of the velocity and the acceleration in the radial and transverse directions are, therefore,

In the case of a particle moving along a circle of center O, have r = constant and

KINEMATICS OF RIGID BODIES Investigate the relations existing between the

time, the positions, the velocities, and the accelerations of the various particles

forming a rigid body.

Various types of rigid-body motion

Translation A motion is said to be a translation if any straight line inside the

body keeps the same direction during the motion.

Rectilinear translation (Paths are straight lines)

Curvilinear translation (Paths are curved lines)

Kinematics of rigid bodies

Rotation about a Fixed Axis Particles forming the rigid body move in parallel planes along circles centered on the same fixed axis called the axis of rotation. The particles located on the axis have zero velocity and zero acceleration

Rotation and the curvilinear translation are not the same.

General Plane Motion Motions in which all the particles of the body move in

parallel planes.

Any plane motion which is neither a rotation nor a translation is referred to as

a general plane motion.

Examples of general plane motion :

Motion about a Fixed Point The three-dimensional motion of a rigid body

attached at a fixed point O, e.g., the motion of a top on a rough floor is known as

motion about a fixed point.

General Motion Any motion of a rigid body which does not fall in any of the

categories above is referred to as a general motion

Translation (either rectilinear or curvilinear translation)

Since A and B, belong to the same rigid body, the derivative of rB/A is zero

When a rigid body is in translation, all the points of the body have the same velocity and the same acceleration at any given instant. In the case of curvilinear translation, the velocity and acceleration change in direction as well as in magnitude at every instant.

Rotation about a fixed axis

Consider a rigid body which rotates about a fixed axis AA’

‘P’ be a point of the body and ‘r’ its position vector with respect to a fixed frame of reference.

The angle θ depends on the position of P within the body, but the rate of change Ѳ is itself independent of P.)

The velocity v of P is a vector perpendicular to the plane containing AA’ and r.

The vector

It is angular velocity of the body and is equal in magnitude to the rate of change of Ѳ with respect to time.

The acceleration ‘a’ of the particle ‘P’

Α is the angular acceleration of a body rotating about a fixed axis is a vector directed along the axis of rotation, and is equal in magnitude to the rate of change of ‘ω’ with respect to time

Two particular cases of rotation

Uniform Rotation This case is characterized by the fact that the angular acceleration is zero. The angular velocity is thus constant.

Uniformly Accelerated Rotation In this case, the angular acceleration is constant

General plane motion The sum of a translation and a rotation

Absolute and relative velocity in plane motion

Any plane motion of a slab can be replaced by a translation defined by the motion of an arbitrary reference point A and a simultaneous rotation about A.

The absolute velocity vB of a particle B of the slab is

The velocity vA corresponds to the translation of the slab with A, while the relative

velocity vB/A is associated with the rotation of the slab about A and is measured with

respect to axes centered at A and of fixed orientation

Consider the rod AB. Assuming that the velocity vA of end A is known, we propose to

find the velocity vB of end B and the angular velocity ω of the rod, in terms of the

velocity vA, the length l, and the angle θ.

The angular velocity ω of a rigid body in plane motion is independent of the reference point.

At any given instant the velocities of the various particles of the slab are the same as if

the slab were rotating about a certain axis perpendicular to the plane of the slab,

called the instantaneous axis of rotation.

Instantaneous Centre

As far as the velocities are concerned, the slab seems to rotate about the instantaneous center C. If vA and vB were parallel and having same magnitude the instantaneous center C would be at an infinite distance and ω would be zero; All points of the slab would have the same velocity.

If vA = 0, point A is itself is the instantaneous center of rotation, and if ω = 0, all the particles have the same velocity vA.

Concept of instantaneous center of rotation

At the instant considered, the velocities of all the particles of the rod are thus the same as if the rod rotated about C.

Thank you.