Physics for Engineers

2. Tension Formula
The tension on an object is equal to the mass of the object x gravitational force
plus/minus the mass x acceleration.
T = mg + ma
T = tension, N, kg-m/s2
m = mass, kg
g = gravitational force, 9.8 m/s2
a = acceleration, m/s2
Tension Formula Questions:
Tension Formula Questions:
1) There is a 5 kg mass hanging from a rope. What is the tension in the rope if
the acceleration of the mass is zero?
Answer: The mass, m = 5 kg; the acceleration, a = 0; and g is defined.
T = mg + ma
T= (5 kg) (9.8 m/s2
) + (5 kg)(0)
T = 49 kg-m
/s2
= 49 N
2) Now assume an acceleration of + 5 m/s2
upwards.
T = mg + ma
T = (5 kg) (9.8 m/s2
) + (5 kg)(5 m/s2
)
T = 49 kg-m
/s2
+ 25 kg-m
/s2
T = 74 kg-m
/s2
= 74 N

3. Rotational Motion of a Rigid Body
Rotational motion is more complicated than linear motion, and only the
motion of rigid bodies will be considered here.
• A rigid body is an object with a mass that holds a rigid shape, such as a
phonograph turntable, in contrast to the sun, which is a ball of gas.
• Many of the equations for the mechanics of rotating objects are similar
to the motion equations for linear motion.
 The angular displacement of a rotating wheel is the angle between the
radius at the beginning and the end of a given time interval. The SI units are
radians.
 The average angular velocity (ω, Greek letter omega), measured in radians
per second, is

4. The angular acceleration (α, Greek letter alpha) has the same form as the linear quantity
and is measured in radians/second/second or rad/s 2.
The kinematics equations for rotational motion at constant angular acceleration are

5. Torque
It is easier to open a door by pushing on the edge farthest from the hinges
than by pushing in the middle. It is intuitive that the magnitude of the force
applied and the distance from the point of application to the hinge affect
the tendency of the door to rotate. This physical quantity, torque, is t = r ×
F sin θ, where F is the force applied, r is the distance from the point of
application to the center of the rotation, and θ is the angle from r to F.

6. Rotational Kinetic Energy Formula
Kinetic energy is the energy of moving objects, including objects that are
rotating. The kinetic energy of a rotating object depends on the object's angular
(rotational) velocity in radians per second, and on the object's moment of
inertia. Moment of inertia is a measure of how easy it is to change the rotation
of an object. Moments of inertia are represented with the letter I, and are
expressed in units of kg∙m2
. The unit of kinetic energy is Joules (J). In terms of
other units, one Joule is equal to one kilogram meter squared per second
squared ( ).
K = kinetic energy ( )
I = moment of inertia (kg∙m2
)
ω = angular velocity (radians/s)

7. Rotational Kinetic Energy Formula Questions:
1) A round mill stone with a moment of inertia of I = 1500 kg∙m2
is rotating at
an angular velocity of 8.00 radians/s. What is the stone's rotational kinetic
energy?
Answer: The rotational kinetic energy of the mill stone can be found using the
formula:
K = 48 000 J
The rotational kinetic energy of the mill stone is 48 000 J.

8. Torque Formula (Moment of Inertia and Angular Acceleration)
• In rotational motion, torque is required to produce an angular
acceleration of an object.
• The amount of torque required to produce an angular acceleration
depends on the distribution of the mass of the object.
• The moment of inertia is a value that describes the distribution.
• It can be found by integrating over the mass of all parts of the object
and their distances to the center of rotation, but it is also possible to
look up the moments of inertia for common shapes.
• The torque on a given axis is the product of the moment of inertia
and the angular acceleration.
• The units of torque are Newton-meters (N∙m).
torque = (moment of inertia)(angular acceleration)
τ = Iα
τ = torque, around a defined axis (N∙m)
I = moment of inertia (kg∙m2)
α = angular acceleration (radians/s2)

9. Torque Formula Questions:
1) The moment of inertia of a solid disc is , where M is the mass of the
disc, and R is the radius. The wheels of a toy car each have a mass of 0.100 kg,
and radius 20.0 cm. If the angular acceleration of a wheel is 1.00 radians/s2
,
what is the torque?
Answer: The torque can be found using the torque formula, and the moment of
inertia of a solid disc. The torque is:
τ = Iα
τ = 0.0020 N∙m
The torque applied to one wheel is 0.0020 N∙m.

10. In physics, elasticity (from Greek
ἐλαστός "ductible") is the ability of a
body to resist a distorting influence or
deforming force and to return to its
original size and shape when that
influence or force is removed. Solid
objects will deform when adequate
forces are applied on them.
In physics and materials
science, plasticity describes the
deformation of a (solid) material
undergoing non-reversible changes of
shape in response to applied forces.
For example, a solid piece of metal
being bent or pounded into a new
shape displays plasticity as permanent
changes occur within the material itself.

11. In general, an elastic modulus is the ratio of stress to strain.
Young's modulus, the bulk modulus, and the shear modulus describe the
response of an object when subjected to tensile, compressional, and shear
stresses, respectively. When an object such as a wire or a rod is subjected to a
tension, the object's length increases.
Young's modulus is defined as the ratio of tensile stress and tensile strain.
Tensile stress is a measure of the deformation that causes stress. Its definition
is the ratio of tensile force (F) and the cross‐sectional area normal to the
direction of the force (A). Units of stress are newtons per square meter (N/m 2).
Tensile strain is defined as the ratio of the change in length ( l o − l) to the
original length ( l o). Strain is a number without units; therefore, the expression
for Young's modulus is

12. If an object of cubic shape has a force applied pushing each face inward, a
compressional stress occurs.
Pressure is defined as force per area P = F/A. The SI unit of pressure is the pascal,
which is equal to 1 newton/meter 2 or N/m 2. Under uniform pressure, the object will
contract, and its fractional change in volume (V) is the compressional strain.
The corresponding elastic modulus is called the bulk modulus and is given by
B = − P/(Δ V/ V o). The negative sign ensures that B is always a positive number
because an increase in pressure causes a decrease in volume.
Applying a force on the top of an object that is parallel to the surface on which it rests
causes a deformation. For example, push the top of a book resting on a tabletop so
that the force is parallel to surface. The cross‐section shape will change from a
rectangle to a parallelogram due to the shear stress (see Figure 1). Shear stress is
defined as the ratio of the tangential force to the area (A) of the face being
stressed. Shear strain is the ratio of the horizontal distance the sheared face moves
(Δ x) and the height of the object (h), which leads to the shear modulus:
Figure 1 Shear stress deforms a book.

14. Hooke's Law Formula Questions:
1) A long, slinky spring has a spring constant of . If the spring is
stretched 1.50 m, what is the restoring force exerted by the spring?
Answer: The restoring force can be found using the formula for Hooke's Law:
The restoring force of the spring is -2.40 N (Newtons). This means the force has
a magnitude of 2.40 N, and is directed toward the equilibrium position.

17. The simple pendulum
The simple pendulum is the idealized model of a mass swinging on the end
of a massless string. For small arcs of swing of less than 15 degrees, the
motion of the pendulum approximates SHM. The period of the pendulum is
given by T = 2π√ l/ g, where l is the length of the pendulum and g is the
acceleration due to gravity. Notice that the period of a pendulum
is not dependent upon the pendulum's mass.
Oscillation is the repetitive variation,
typically in time, of some measure
about a central value (often a point
of equilibrium) or between two or
more different states. The
term vibration is precisely used to
describe mechanical oscillation.
Familiar examples of oscillation
include a
swinging pendulum and alternating
current power.

20. PREPARED BY:
ENGR. JAY GABRIEL F. JIMENEZ
M.Eng.,REE,RME