TORQUE, CENTER OF MASS, CENTER OF GRAVITY
Motion in which an entire object moves is called translation
Motion in which an object spins is called rotation
The point or line about which an object turns is its center of rotation
The LEVER ARM, d, is the
perpendicular distance from the
axis of rotation to the line of
action of the force.
d = L sin Φ
FULCRUM - the point of support,
or axis, about which a lever may be made to rotate
How does force create rotation ??
A torque is an action that causes objects
Torque is not the same thing as force.
TORQUE is the cross product between the distance vector
(the distance from the pivot point to the point where force is applied)
and the force vector
magnitude : rF sin
To make an object rotate, a force must be applied in
the right place.
magnitude : rF sin
Consider force required to open door.
Is it easier to open the door by pushing/pulling
away from hinge or close to hinge?
The amount of torque depends on where and in what direction the
force is applied, as well as the location of the axis of rotation.
of the applied
force causes the
door to open?
Using the concept of torque, explain why the “easy-off” cap is easier
to unscrew than the normal cap.
Some Physics Quantities
Vector - quantity with both magnitude (size) and direction
Scalar - quantity with magnitude only
• Force •Work
The cross product of two vectors a and b is defined as
c ab rF
whose magnitude is
c ab sin rF sin
where is the angle (< 180o) between a and b,
whose direction is perpendicular to both a and b
in the sense of the right-hand rule.
Cross product (right-hand rule )
c ab Use the right-hand rule to
determine the direction of the
resulting vector in a cross
product. Hold your right hand in
front of you so that the thumb is
pointed up, the index finger is
pointed away from you and the
middle finger is pointed to your
left. The index finger shows the
direction of vector A, the middle
finger shows the direction of
vector B and the thumb shows
the direction of the vector from
the cross product A x B.
• A torque ( a vector quantity) that tends o produce
a counterclockwise rotation is considered positive.
• A torque that tends to produce a clockwise rotation
By convention, the sign of torque is:
A force of 50 newtons is applied to a wrench that is 30
centimeters long. Calculate the torque if the force is applied
perpendicular to the wrench so the lever arm is 30 cm
rF sin (0.3m)(50 N ) sin 90 o
rF sin (0.3m)(50m)(1) 15Nm
For the same force, you get more torque with a bigger
wrench the job is easier!
The torque can be increased by applying the force at right angles to
the lever arm or by extending the lever arm.
For your arm, leg or any body part to move the appropriate
muscles and bones must work together as a series of levers.
A lever comprises of three components :
Fulcrum or pivot - the point about which the lever rotates
Load - the force applied by the lever system
Effort - the force applied by the user of the lever system
The way in which a lever will operate is dependent on the type of
Levers are important in human
motion because the human
body is a system of levers.
Our joints are axes of rotation
(fulcrums) and our bones are
the levers. Forces to move the
levers are provided by our
muscles, gravity, and external
Three Classes of Levers
First Class - fulcrum
between Input and output
Second Class – output
between fulcrum and input
Third Class – input
between fulcrum and
First Class Second Class
• To ensure that an object
does not accelerate or
rotate two conditions
must be met:
• net force = 0
• net torque = 0
• First Condition of Equilibrium
• The net external force must be zero
Fx 0 and Fy 0
– This is a statement of translational equilibrium
• Second Condition of Equilibrium
• The net external torque must be zero
• This is a statement of rotational equilibrium
For equilibrium to exists, torques must add to zero.
The torques applied by the 2 kids are equal and opposite,
so the see-saw doesn’t move.
Balanced torques happen on a seesaw when two children
of different weights sit at different positions
We know that the heavier kid needs to sit
closer to the center of the see-saw. Why?
In order for the see-saw to balance, there
must be no rotation. This means that
there cannot be any unbalanced torques
acting on the see-saw.
The heavier kid has more weight (force) so
must have a smaller lever arm in order to
make the product of force and lever arm
(torque) equal that of the smaller kid sitting
further from the axis.
• Direction of rotation of applied torque is very important (i.e.
clockwise or anticlockwise).
• Torques can add or oppose each other.
• If two opposing torques are of equal magnitude they will cancel one
another to create a balanced system.
W1 = m1.g W2 = m2.g
Torque = Fr
W1.r1 = W2.r2
or m1.g.r1 = m2. g.r2
Thus at balance: m1.r1 = m2.r2
(This is the principle of weighing scales.)
If an object is supported at one
special point, it will balance.
This point is called the center of
•For SYMMETRICAL objects,
such as a meter stick, the CG is
located at the actual center.
• For ASYMMETRICAL objects,
such as a hammer, the CG is
located away from the center,
closer to the hammer head (the
Center of Mass (CM)
• An object can be divided into
many small particles
– Each particle will have a
specific mass and specific
• The x coordinate of the center
of mass will be
m x i i
• Similar expressions can be
found for the y and z
A system can often be well represented
as though its mass were concentrated
at single point,
the center of mass
Center of Mass examples
CM for the Human Body
The location of the center of mass
of the leg (circled) will depend
on the position of the leg.
High jumpers have developed a technique
where their CM actually passes under the bar
as they go over it. This allows them to clear
Center of Gravity
• All the various gravitational
forces acting on all the
various mass elements are
equivalent to a single
gravitational force acting
through a single point called
the center of gravity.
The center of gravity is the point where the gravitational force can
be considered to act. It is the same as the center of mass as long
as the gravitational force does not vary among different parts of
A force applied at the CG of an object results in straight-line
acceleration, but no rotation.
A force applied away from the CG will have a “lever
arm”, which produces a torque. The result is straight-line
acceleration and rotation.
Why things fall over
• Every object has a special point called the
center of gravity (CG).
• if the center of gravity is supported, the object
will not fall over.
Touch your toes
while standing against a wall
Why things fall over