November 2012

819 Inverted T-section | Moment of Inertia

Problem 819
Determine the moment of inertia of the T-section shown in Fig. P-819 with respect to its centroidal Xo axis.
 

820 Unsymmetrical I-section | Moment of Inertia

Problem 820
Determine the moment of inertia of the area shown in Fig. P-819 with respect to its centroidal axes.
 

Dynamics

Dynamics is the branch of mechanics which deals with the study of bodies in motion.
 

Branches of Dynamics
Dynamics is divided into two branches called kinematics and kinetics.

Kinematics is the geometry in motion. This term is used to define the motion of a particle or body without consideration of the forces causing the motion.

Kinetics is the branch of mechanics that relates the force acting on a body to its mass and acceleration.
 

Symbols and Notations
s = distance

Kinematics

Motion of a Particle
Particle is a term used to denote an object of point size. A system of particles which formed into appreciable size is termed as body. These terms may apply equally to the same object. The earth for example may be assumed as a particle in comparison with its orbit, whereas to an observer on the earth, it is a body with appreciable size. In general, a particle is an object whose size is so small in comparison to the size of its path.
 

Rectilinear Translation (Motion Along a Straight Line)

Curvilinear Translation (Projectile Motion)

Projectile motion follows a parabolic trajectory. The vertical component of projectile is under constant gravitational acceleration and the horizontal component is at constant velocity. For easy handling, resolve the motion into x and y components and use the formulas in rectilinear translation.
 

Form the figure below:

$v_{ox} = v_o \, \cos \theta$

$v_{oy} = v_o \, \sin \theta$

 

000-projectile-general.gif

 

1002 Location of warning torpedo | Rectilinear Translation

Problem 1002
On a certain stretch of track, trains run at 60 mph (96.56 kph). How far back of a stopped train should be a warning torpedo be placed to signal an oncoming train? Assume that the brakes are applied at once and retard the train at the uniform rate of 2 ft/sec2 (0.61 m/s2).
 

1003 Return in 10 seconds | Rectilinear Translation

Problem 1003
A stone is thrown vertically upward and return to earth in 10 sec. What was its initial velocity and how high did it go?
 

1004 Relative velocity | Rectilinear Translation

Problem 1004
A ball is dropped from the top of a tower 80 ft (24.38 m) high at the same instant that a second ball is thrown upward from the ground with an initial velocity of 40 ft/sec (12.19 m/s). When and where do they pass, and with what relative velocity?
 

1005 Finding the depth of well by dropping a stone | Rectilinear Translation

Problem 1005
A stone is dropped down a well and 5 sec later, the sounds of the splash is heard. If the velocity of sound is 1120 ft/sec (341.376 m/s), what is the depth of the well?
 

1007 Finding when and where the stones pass each other | Rectilinear Translation

Problem 1007
A stone is dropped from a captive balloon at an elevation of 1000 ft (304.8 m). Two seconds later another stone is thrown vertically upward from the ground with a velocity of 248 ft/s (75.6 m/s). If g = 32 ft/s2 (9.75 m/s2), when and where the stones pass each other?
 

1008 Stones thrown vertically upward | Rectilinear Translation

Problem 1008
A stone is thrown vertically upward from the ground with a velocity of 48.3 ft per sec (14.72 m per sec). One second later another stone is thrown vertically upward with a velocity of 96.6 ft per sec (29.44 m per sec). How far above the ground will the stones be at the same level?
 

1019 Velocity and acceleration from the equation of distance | Motion with Variable Acceleration

Problem 1019
The motion of a particle is given by the equation $s = 2t^4 - \frac{1}{6}t^3 + 2t^2$ where $s$ is in meter and $t$ in seconds. Compute the values of $v$ and $a$ when $t = 2 \, \text{ sec}$.
 

1009 Initial velocity of the second ball | Rectilinear Translation

Problem 1009
A ball is shot vertically into the air at a velocity of 193.2 ft per sec (58.9 m per sec). After 4 sec, another ball is shot vertically into the air. What initial velocity must the second ball have in order to meet the first ball 386.4 ft (117.8 m) from the ground?
 

1010 Time to wait in dropping a stone | Rectilinear Translation

Problem 1010
A stone is thrown vertically up from the ground with a velocity of 300 ft per sec (91.44 m/s). How long must one wait before dropping a second stone from the top of a 600-ft (182.88-m) tower if the two stones are to pass each other 200 ft (60.96 m) from the top of the tower?
 

1011 Time of launching a ship | Rectilinear Translation

Problem 1011
A ship being launched slides down the ways with constant acceleration. She takes 8 sec to slide (the first foot | 0.3048 meter). How long will she take to slide down the ways if their length is (625 ft | 190.5 m)?
 

Problem 10 | Special Products and Factoring

Problem 10
Given that $x + y + xy = 1$, where $x$ and $y$ are nonzero real numbers,find the value of $xy + \dfrac{1}{xy} - \dfrac{y}{x} - \dfrac{x}{y}$.
 

10 Solving for angle A in triangle ABC

Problem 10
In a triangle ABC, if   $\dfrac{2 \cos A}{a} + \dfrac{\cos B}{b} + \dfrac{2 \cos C}{c} = \dfrac{a}{bc} + \dfrac{b}{ca}$,   find the value of angle $A$.
 

02 Problem involving angle and median | Properties of a Triangle

Problem 02
From the figure shown below, angle CAD = angle BCD = theta and CD is a median of triangle ABC through vertex C. Determine the value of the angle theta.
 

02-median-angle-theta.gif

 

1012 Train at constant deceleration | Rectilinear Translation

Problem 1012
A train moving with constant acceleration travels 24 ft (7.32 m) during the 10th sec of its motion and 18 ft (5.49 m) during the 12th sec of its motion. Find its initial velocity and its constant acceleration.
 

Strain Energy Method (Castigliano’s Theorem) | Beam Deflection

Alberto Castigliano (Catigliano's Theorem)
Engr. Alberto Castigliano

Italian engineer Alberto Castigliano (1847 – 1884) developed a method of determining deflection of structures by strain energy method. His Theorem of the Derivatives of Internal Work of Deformation extended its application to the calculation of relative rotations and displacements between points in the structure and to the study of beams in flexure.
 

Conjugate Beam Method | Beam Deflection

Slope on real beam = Shear on conjugate beam
Deflection on real beam = Moment on conjugate beam

 

Properties of Conjugate Beam

Otto Mohr
Engr. Christian Otto Mohr
  1. The length of a conjugate beam is always equal to the length of the actual beam.
  2. The load on the conjugate beam is the M/EI diagram of the loads on the actual beam.

01 - Highest point of projectile as measured from inclined plane

Problem 01
A projectile is fired up the inclined plane at an initial velocity of 15 m/s. The plane is making an angle of 30° from the horizontal. If the projectile was fired at 30° from the incline, compute the maximum height z measured perpendicular to the incline that is reached by the projectile. Neglect air resistance.
 

Problem 653 | Beam Deflection by Conjugate Beam Method

Problem 653
Compute the midspan value of EIδ for the beam shown in Fig. P-653. (Hint: Draw the M diagram by parts, starting from midspan toward the ends. Also take advantage of symmetry.
 

653-conjugate-beam-method.gif

 

Problem 654 | Beam Deflection by Conjugate Beam Method

Problem 654
For the beam in Fig. P-654, find the value of EIδ at 2 ft from R2.
 

654-conjugate-beam-method.gif