Elementary dynamics - Baker
ELEMENTARY DYNAMICS
BY W. M. BAKER,
HEAD MASTER OF THE MILITARY AND CIVIL DEPARTMENT AT CHELTENHAM COLLEGE.
LONDON; CAMBRIDGE; DEIGHTON, BELL AND CO.; 1905
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PREFACE TO THE SECOND EDITION
The main points of this edition are:
1. The insertion of geometrical and graphical proofs and representations of the formulae
V = u + ft, s = ut + 1/2ft2, v2 = u2 + 2fs
2. Velocity treated as the limiting value of Ds/Dt
3. Acceleration treated as the limiting value of Dv/Dt
4. The treatment of Work and Energy at an earlier stage
5. The introduction of batches of Revision Papers at different stages
6. Full geometrical and graphical treatment of Projectiles.
7. The introduction of worked-out problem of various types, e.g.
- Trail of smoke problems,
- Problems on Change of Velocity,
- Jerks of Inelastic Strings,
- Motion on a Moving Inclined Plane
8. Motion of a Particle under a variable acceleration.
Altogether about sixty pages of new matter have been inserted.
The recommendations of the Committee of the Mathematical Association have been followed to a great extent, especially in leading up to the ideas of the Differential and Integral Calculus.
The student is assumed to possess a knowledge of elementary Trigonometry, Graphs, and in the later chapters, of the Parabola, and elementary Co-ordinate Geometry.
In some sets of examples, the harder problems are marked with an asterisk. An endeavour has been made to put all explanations in as simple a manner as possible for the sake of beginners, but at the same time, the range of work should be sufficient for Woolwich and Sandhurst candidates, and for students reading for Scholarships at the Universities.
With the kind permission of Mr J. M. Dyer, of Eton, I have inserted a good many examples from the collection (Mathematical Examples, now out of print) made by him and the late Mr R. Prowde Smith.
1. The insertion of geometrical and graphical proofs and representations of the formulae
V = u + ft, s = ut + 1/2ft2, v2 = u2 + 2fs
2. Velocity treated as the limiting value of Ds/Dt
3. Acceleration treated as the limiting value of Dv/Dt
4. The treatment of Work and Energy at an earlier stage
5. The introduction of batches of Revision Papers at different stages
6. Full geometrical and graphical treatment of Projectiles.
7. The introduction of worked-out problem of various types, e.g.
- Trail of smoke problems,
- Problems on Change of Velocity,
- Jerks of Inelastic Strings,
- Motion on a Moving Inclined Plane
8. Motion of a Particle under a variable acceleration.
Altogether about sixty pages of new matter have been inserted.
The recommendations of the Committee of the Mathematical Association have been followed to a great extent, especially in leading up to the ideas of the Differential and Integral Calculus.
The student is assumed to possess a knowledge of elementary Trigonometry, Graphs, and in the later chapters, of the Parabola, and elementary Co-ordinate Geometry.
In some sets of examples, the harder problems are marked with an asterisk. An endeavour has been made to put all explanations in as simple a manner as possible for the sake of beginners, but at the same time, the range of work should be sufficient for Woolwich and Sandhurst candidates, and for students reading for Scholarships at the Universities.
With the kind permission of Mr J. M. Dyer, of Eton, I have inserted a good many examples from the collection (Mathematical Examples, now out of print) made by him and the late Mr R. Prowde Smith.
CONTENTS.
Chapter I.
- Velocity. Uniform Motion
Chapter II.
- Uniform Acceleration
- Graphical proofs and treatment of the formulae
V = u + ft, s = ut + 1/2ft2, v2 = u2 + 2fs
- Limiting value of Ds/Dt = velocity
- Limiting value of Dv/Dt = acceleration
- Velocity at any point on a space-time curve
Chapter III.
- Vertical Motion under Gravity
Chapter IV.
- Laws of Motion. P = mf. Absolute unit of force
Chapter V.
- Parallelogram of Velocities
- Triangle of Velocities
- Polygon of Velocities
Chapter VI.
- Parallelogram of Accelerations
- Parallelogram of Forces
- Motion on a smooth inclined plane
- Line of Quickest Descent
Chapter VII.
- Masses connected by strings passing over pulleys
- Atwood's Machine
Chapter VIII.
- Bodies on moving horizontal planes, floors of Lifts, etc
- Motion on rough planes
Chapter IX.
- Revision Questions
- Revision Papers
- Miscellaneous Examples
Chapter X.
- Work. Rate of doing work Efficiency of a machine
- Graphical representation of work done
- Energy
Chapter XI.
- Relative Motion. Harder Pulley Problems
- Trail of Smoke Problems
- Motion on a moving inclined plane
- Angular Velocity
- Change of Velocity
Chapter XII.
- Uniform Circular Motion
- Conical Pendulum
- Stress at any point of a revolving ring
Chapter XIII.
- Projectiles. Horizontal Range, Time of flight, etc
- Graphical representation of the path of a projectile
- Graphical determination of the velocity at any point
Chapter XIV.
- Projectiles. Range on an inclined plane
- Maximum range
- Proof that the path of a projectile is a parabola
- Various geometrical problems concerning projectiles
Chapter XV.
- Revision Papers
Chapter XVI.
- Impulse and Collision
- Motion of Shot and Gun
- Elasticity
- Impact on a fixed plane
- Direct impact of smooth balls
- Oblique impact of smooth balls
- Action during Impact
- Jerks of Inelastic Strings
- Problem on continued impacts
- Problem on rain pressure
- Problem on pressure of a jet of water
- Problem on pile driving
- Impact on a rough plane
- Loss of Energy due to Impact
- Penetration of shot into iron plate
- Pressure of a heavy falling chain on a table
Chapter XVII.
- Motion on a smooth curve under the action of gravity
Chapter XVIII.
- Simple Harmonic Motion
- Pendulum
- Variation in the value of
Chapter XIX.
- Motion of Centre of Gravity
Chapter XX.
- Units
Chapter XXI.
- Initial actions, Tension, Motion, etc
Chapter XXII.
- Motion of a particle moving under a variable acceleration
Chapter XXIII.
- Miscellaneous Examples
CHAPTER I - VELOCITY
A point is said to be in motion when it changes its position relative to surrounding objects.
Thus if the distance from a point B to a point A changes, B is said to have motion relative to A ; and we must notice that this change may be a change in direction as well as a hange in lengiL At present we shall only consider a change a length.
Def. The velocity of a moving point is its rate of motion; or the rate at which it is changing its position.
The velocity of a point is said to be uniform when the point moves over equal distances in equal intervals of time, however small those intervals may be.
Measurement of velocity. When uniform the velocity of a point is measured by the distance passed over in a unit of time; when variable, it is measured, by the distance which would be passed over in unit time if the point moved during that unit of time with the same velocity as at the instant under consideration.
The unit of time usually employed is one second. The British standard unit of length is a yard, which is defined as the distance between the centres of two plugs in a bronze bar kept in the Exchequer Office, the temperature
of the bar being 62 degrees Fahrenheit.
The unit generally employed by engineers in England is one-third of this distance, and is called a foot.
For scientific purposes the unit of length usually employed is a centimetre.
The unit of velocity is the velocity of a point which moves uniformly over unit space in unit time.
Thus a point which passes uniformly over one foot in one second has unit velocity when a foot and a second are taken as units of length and time respectively. Also when we speak of a point as having a velocity v, we mean that it passes over v units of length in unit time; and a point having an equal velocity but in an opposite direction has a velocity - v.
5. When a point moves with uniform velocity u, it passes over u units of length in unit time, therefore it passes over ut units of length in t units of time. Hence if 5 be the number of units of length passed over by a point moving with velocity u for t units of time,
s = u * t
The mean or average velocity of a moving point during any interval (in which it is not moving uniformly), is the velocity of another point which, moving uniformly, passes over the same distance in the same time.
CHAPTER II - ACCELERATION.
Def. The acceleration of a moving point is the rate of change of its velocity.
When uniform acceleration is measured by the change of velocity in unit time.
When variable acceleration is measured at any instant by the change of velocity which would take place in unit time if the acceleration remained during that time the same as at the instant under consideration.
At present we shall only deal with uniform accelerations.
A point is said to move with unit acceleration when its velocity is changed by the unit of velocity in each unit of time.
Thus if ft.-sec. units be used, a point moves with unit acceleration when its velocity is increased by one foot per second during every second.
Or again, if a body has an acceleration 6 ft.-sec. units we mean that in every second of its motion its velocity is increased by 6 ft. per second.
A point is said to have a negative acceleration when its velocity is decreasing. A negative acceleration is therefore the same as a retardation.
A point, starting with velocity u, moves in a straight line subject to a constant acceleration f in its direction of motion: if v be its velocity at time t, to prove that
V = u + ft
By the definition of acceleration, f denotes the change of velocity in each unit of time; therefore ft denotes the change in t units of time.
Thus if the distance from a point B to a point A changes, B is said to have motion relative to A ; and we must notice that this change may be a change in direction as well as a hange in lengiL At present we shall only consider a change a length.
Def. The velocity of a moving point is its rate of motion; or the rate at which it is changing its position.
The velocity of a point is said to be uniform when the point moves over equal distances in equal intervals of time, however small those intervals may be.
Measurement of velocity. When uniform the velocity of a point is measured by the distance passed over in a unit of time; when variable, it is measured, by the distance which would be passed over in unit time if the point moved during that unit of time with the same velocity as at the instant under consideration.
The unit of time usually employed is one second. The British standard unit of length is a yard, which is defined as the distance between the centres of two plugs in a bronze bar kept in the Exchequer Office, the temperature
of the bar being 62 degrees Fahrenheit.
The unit generally employed by engineers in England is one-third of this distance, and is called a foot.
For scientific purposes the unit of length usually employed is a centimetre.
The unit of velocity is the velocity of a point which moves uniformly over unit space in unit time.
Thus a point which passes uniformly over one foot in one second has unit velocity when a foot and a second are taken as units of length and time respectively. Also when we speak of a point as having a velocity v, we mean that it passes over v units of length in unit time; and a point having an equal velocity but in an opposite direction has a velocity - v.
5. When a point moves with uniform velocity u, it passes over u units of length in unit time, therefore it passes over ut units of length in t units of time. Hence if 5 be the number of units of length passed over by a point moving with velocity u for t units of time,
s = u * t
The mean or average velocity of a moving point during any interval (in which it is not moving uniformly), is the velocity of another point which, moving uniformly, passes over the same distance in the same time.
CHAPTER II - ACCELERATION.
Def. The acceleration of a moving point is the rate of change of its velocity.
When uniform acceleration is measured by the change of velocity in unit time.
When variable acceleration is measured at any instant by the change of velocity which would take place in unit time if the acceleration remained during that time the same as at the instant under consideration.
At present we shall only deal with uniform accelerations.
A point is said to move with unit acceleration when its velocity is changed by the unit of velocity in each unit of time.
Thus if ft.-sec. units be used, a point moves with unit acceleration when its velocity is increased by one foot per second during every second.
Or again, if a body has an acceleration 6 ft.-sec. units we mean that in every second of its motion its velocity is increased by 6 ft. per second.
A point is said to have a negative acceleration when its velocity is decreasing. A negative acceleration is therefore the same as a retardation.
A point, starting with velocity u, moves in a straight line subject to a constant acceleration f in its direction of motion: if v be its velocity at time t, to prove that
V = u + ft
By the definition of acceleration, f denotes the change of velocity in each unit of time; therefore ft denotes the change in t units of time.
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