Short Notes: Impulse and Momentum | Short Notes for Mechanical Engineering PDF Download

 Page 1


Impulse and Momentum
Linear Momentum and its Conservation
The linear momentum of a particle of mass m moving with velocity "v is defined as
— * — t
p = IHV
• Linear momentum, being the product of a scalar and a vector, is a vector.
• It has dimensions M.L.T-1 and units kg.m.s-1.
• The direction of the momentum vector is the same as the velocity vector.
We can express Newton’s second law in terms of linear momentum in this way
V r dP 
" d! di
dv
— m ¦ — = m . 
d!
• Thus the resultant force on an object (system) equals the time rate of change 
of linear momentum of the object (system).
• The definition of linear momentum enables us to put the second law into a 
more general powerful form.
• If, in addition, the system is an isolated one then we can formulate a law of 
conservation of linear momentum.
Impulse and Momentum
• Impulse is defined as simply change in momentum. In a collision between two 
particles (and especially a contact collision) the force
of interaction might vary with time.
Page 2


Impulse and Momentum
Linear Momentum and its Conservation
The linear momentum of a particle of mass m moving with velocity "v is defined as
— * — t
p = IHV
• Linear momentum, being the product of a scalar and a vector, is a vector.
• It has dimensions M.L.T-1 and units kg.m.s-1.
• The direction of the momentum vector is the same as the velocity vector.
We can express Newton’s second law in terms of linear momentum in this way
V r dP 
" d! di
dv
— m ¦ — = m . 
d!
• Thus the resultant force on an object (system) equals the time rate of change 
of linear momentum of the object (system).
• The definition of linear momentum enables us to put the second law into a 
more general powerful form.
• If, in addition, the system is an isolated one then we can formulate a law of 
conservation of linear momentum.
Impulse and Momentum
• Impulse is defined as simply change in momentum. In a collision between two 
particles (and especially a contact collision) the force
of interaction might vary with time.
The force is relatively short-lived, being zero before clock time ti and zero 
after clock time tf and having a relatively large value at maximum.
The elapsed time for the interaction is to a good approximation At = tf - ti.
A
fig above shows A force that varies over a relatively short elapsed time. The area 
under the force curve is equal to the magnitude of the impulse
• The sum of forces is a function of time.
• To find the change in momentum we must integrate over the elapsed time for 
the interaction:
• The change in momentum is defined as the impulse and given the symbol "J
• Impulse has the same dimensions and units as momentum and is also a 
vector.
• The impulse has a magnitude equal to the area under the force curve between 
the two clock times, that is, over the elapsed time of the collision.
• The direction of the impulse vector is the same as the direction of the change 
in momentum vector.
By Newton's Second Law, the net force is equal to the mass of an object times its 
acceleration,
F = ma
This can be substituted into the equation for impulse,
ap - A a -
J « J ^ F (t)d t - Ajj.
Page 3


Impulse and Momentum
Linear Momentum and its Conservation
The linear momentum of a particle of mass m moving with velocity "v is defined as
— * — t
p = IHV
• Linear momentum, being the product of a scalar and a vector, is a vector.
• It has dimensions M.L.T-1 and units kg.m.s-1.
• The direction of the momentum vector is the same as the velocity vector.
We can express Newton’s second law in terms of linear momentum in this way
V r dP 
" d! di
dv
— m ¦ — = m . 
d!
• Thus the resultant force on an object (system) equals the time rate of change 
of linear momentum of the object (system).
• The definition of linear momentum enables us to put the second law into a 
more general powerful form.
• If, in addition, the system is an isolated one then we can formulate a law of 
conservation of linear momentum.
Impulse and Momentum
• Impulse is defined as simply change in momentum. In a collision between two 
particles (and especially a contact collision) the force
of interaction might vary with time.
The force is relatively short-lived, being zero before clock time ti and zero 
after clock time tf and having a relatively large value at maximum.
The elapsed time for the interaction is to a good approximation At = tf - ti.
A
fig above shows A force that varies over a relatively short elapsed time. The area 
under the force curve is equal to the magnitude of the impulse
• The sum of forces is a function of time.
• To find the change in momentum we must integrate over the elapsed time for 
the interaction:
• The change in momentum is defined as the impulse and given the symbol "J
• Impulse has the same dimensions and units as momentum and is also a 
vector.
• The impulse has a magnitude equal to the area under the force curve between 
the two clock times, that is, over the elapsed time of the collision.
• The direction of the impulse vector is the same as the direction of the change 
in momentum vector.
By Newton's Second Law, the net force is equal to the mass of an object times its 
acceleration,
F = ma
This can be substituted into the equation for impulse,
ap - A a -
J « J ^ F (t)d t - Ajj.
¦ ¦ ¦ j = m i v
The change in velocity is the difference between the velocities at the starting and 
ending times,
At? =
The formula for impulse becomes,
J = m(v^-v^)
-t __» __»
¦ ¦ ¦ / = rau2 — tiifj 
/ = Pz ~ Pi
This equation is called the impulse-momentum theorem. In words, it states that the 
change in momentum of an object in a certain time interval is equal to the impulse 
of the net force that acts on the object in the time interval. Using this formula, it is 
possible to relate changes in momentum to the forces that were applied to cause 
the change. It also shows that the time over which a force is applied has an effect 
on the change in momentum that results.
It is also important to note that the units for momentum and impulse are effectively 
the same. The unit of momentum is
kg ¦ m/s
, and the unit of impulse is the Newton-second,
N ¦ 5
. The Newton is a compound unit, defined as,
Conservation of Momentum
• When two objects interact, such as in a collision, they may exert forces on 
each other.
• The forces the objects exert on each other can be considered part of
a closed or isolated system. In this case, the forces involved are internal 
forces.
• If any outside forces affect the system, these are called external forces.
• According to Newton's Third Law, when there are no external forces, the 
internal forces that act between two objects have equal magnitudes and 
opposite directions. If the two objects are labelled A and B, the forces they 
exert on each other are,
on B on A
• During a collision, these forces act for the same amount of time. If the 
collision begins at time ti and ends at time X 2, then the time duration of the 
collision is
A t
, and the impulse experienced by object A is,
— » -t
h = F E o n A ^ •
• The impulse experienced by object B is,
Page 4


Impulse and Momentum
Linear Momentum and its Conservation
The linear momentum of a particle of mass m moving with velocity "v is defined as
— * — t
p = IHV
• Linear momentum, being the product of a scalar and a vector, is a vector.
• It has dimensions M.L.T-1 and units kg.m.s-1.
• The direction of the momentum vector is the same as the velocity vector.
We can express Newton’s second law in terms of linear momentum in this way
V r dP 
" d! di
dv
— m ¦ — = m . 
d!
• Thus the resultant force on an object (system) equals the time rate of change 
of linear momentum of the object (system).
• The definition of linear momentum enables us to put the second law into a 
more general powerful form.
• If, in addition, the system is an isolated one then we can formulate a law of 
conservation of linear momentum.
Impulse and Momentum
• Impulse is defined as simply change in momentum. In a collision between two 
particles (and especially a contact collision) the force
of interaction might vary with time.
The force is relatively short-lived, being zero before clock time ti and zero 
after clock time tf and having a relatively large value at maximum.
The elapsed time for the interaction is to a good approximation At = tf - ti.
A
fig above shows A force that varies over a relatively short elapsed time. The area 
under the force curve is equal to the magnitude of the impulse
• The sum of forces is a function of time.
• To find the change in momentum we must integrate over the elapsed time for 
the interaction:
• The change in momentum is defined as the impulse and given the symbol "J
• Impulse has the same dimensions and units as momentum and is also a 
vector.
• The impulse has a magnitude equal to the area under the force curve between 
the two clock times, that is, over the elapsed time of the collision.
• The direction of the impulse vector is the same as the direction of the change 
in momentum vector.
By Newton's Second Law, the net force is equal to the mass of an object times its 
acceleration,
F = ma
This can be substituted into the equation for impulse,
ap - A a -
J « J ^ F (t)d t - Ajj.
¦ ¦ ¦ j = m i v
The change in velocity is the difference between the velocities at the starting and 
ending times,
At? =
The formula for impulse becomes,
J = m(v^-v^)
-t __» __»
¦ ¦ ¦ / = rau2 — tiifj 
/ = Pz ~ Pi
This equation is called the impulse-momentum theorem. In words, it states that the 
change in momentum of an object in a certain time interval is equal to the impulse 
of the net force that acts on the object in the time interval. Using this formula, it is 
possible to relate changes in momentum to the forces that were applied to cause 
the change. It also shows that the time over which a force is applied has an effect 
on the change in momentum that results.
It is also important to note that the units for momentum and impulse are effectively 
the same. The unit of momentum is
kg ¦ m/s
, and the unit of impulse is the Newton-second,
N ¦ 5
. The Newton is a compound unit, defined as,
Conservation of Momentum
• When two objects interact, such as in a collision, they may exert forces on 
each other.
• The forces the objects exert on each other can be considered part of
a closed or isolated system. In this case, the forces involved are internal 
forces.
• If any outside forces affect the system, these are called external forces.
• According to Newton's Third Law, when there are no external forces, the 
internal forces that act between two objects have equal magnitudes and 
opposite directions. If the two objects are labelled A and B, the forces they 
exert on each other are,
on B on A
• During a collision, these forces act for the same amount of time. If the 
collision begins at time ti and ends at time X 2, then the time duration of the 
collision is
A t
, and the impulse experienced by object A is,
— » -t
h = F E o n A ^ •
• The impulse experienced by object B is,
1 b ~ F a onB&t
• If the values for the forces in these impulse equations are substituted in to 
the equation for Newton's Third Law, the result is,
^A a n B ^ B on A
. / h _ J a 
At i t
By the impulse-momentum theorem, this is equivalent to,
Pb , 2 ~ P b , 1 = ~ (P a ,2 ~ P a .l )
• In this equation,
pH
means the momentum of object A at time ti,
Pa ,2
means the momentum of object A at time X 2,
Pb, 1
means the momentum of object B at time ti, and
Pb, 2
means the momentum of object B at time X 2.
• The equation can be rearranged to put all of the terms for time ti on one side, 
and terms for time X 2 on the other,
Pa, 1 T Pb, i = Pa,2 + Pb,2
• In this case, in which there were no external forces, the sum of the momenta 
before the collision is equal to the sum of the momenta after.
• In general, as long as there are no external forces, the total momentum of the 
system is constant. This is known as conservation of momentum.
• For any number of objects the total momentum can be labeled
p
P = P a + P b + ¦ " = ™ A V a + m B V B + ¦ "
If there are no external forces, the total momentum
p
remains constant, even if the momenta of the individual objects change.
Page 5


Impulse and Momentum
Linear Momentum and its Conservation
The linear momentum of a particle of mass m moving with velocity "v is defined as
— * — t
p = IHV
• Linear momentum, being the product of a scalar and a vector, is a vector.
• It has dimensions M.L.T-1 and units kg.m.s-1.
• The direction of the momentum vector is the same as the velocity vector.
We can express Newton’s second law in terms of linear momentum in this way
V r dP 
" d! di
dv
— m ¦ — = m . 
d!
• Thus the resultant force on an object (system) equals the time rate of change 
of linear momentum of the object (system).
• The definition of linear momentum enables us to put the second law into a 
more general powerful form.
• If, in addition, the system is an isolated one then we can formulate a law of 
conservation of linear momentum.
Impulse and Momentum
• Impulse is defined as simply change in momentum. In a collision between two 
particles (and especially a contact collision) the force
of interaction might vary with time.
The force is relatively short-lived, being zero before clock time ti and zero 
after clock time tf and having a relatively large value at maximum.
The elapsed time for the interaction is to a good approximation At = tf - ti.
A
fig above shows A force that varies over a relatively short elapsed time. The area 
under the force curve is equal to the magnitude of the impulse
• The sum of forces is a function of time.
• To find the change in momentum we must integrate over the elapsed time for 
the interaction:
• The change in momentum is defined as the impulse and given the symbol "J
• Impulse has the same dimensions and units as momentum and is also a 
vector.
• The impulse has a magnitude equal to the area under the force curve between 
the two clock times, that is, over the elapsed time of the collision.
• The direction of the impulse vector is the same as the direction of the change 
in momentum vector.
By Newton's Second Law, the net force is equal to the mass of an object times its 
acceleration,
F = ma
This can be substituted into the equation for impulse,
ap - A a -
J « J ^ F (t)d t - Ajj.
¦ ¦ ¦ j = m i v
The change in velocity is the difference between the velocities at the starting and 
ending times,
At? =
The formula for impulse becomes,
J = m(v^-v^)
-t __» __»
¦ ¦ ¦ / = rau2 — tiifj 
/ = Pz ~ Pi
This equation is called the impulse-momentum theorem. In words, it states that the 
change in momentum of an object in a certain time interval is equal to the impulse 
of the net force that acts on the object in the time interval. Using this formula, it is 
possible to relate changes in momentum to the forces that were applied to cause 
the change. It also shows that the time over which a force is applied has an effect 
on the change in momentum that results.
It is also important to note that the units for momentum and impulse are effectively 
the same. The unit of momentum is
kg ¦ m/s
, and the unit of impulse is the Newton-second,
N ¦ 5
. The Newton is a compound unit, defined as,
Conservation of Momentum
• When two objects interact, such as in a collision, they may exert forces on 
each other.
• The forces the objects exert on each other can be considered part of
a closed or isolated system. In this case, the forces involved are internal 
forces.
• If any outside forces affect the system, these are called external forces.
• According to Newton's Third Law, when there are no external forces, the 
internal forces that act between two objects have equal magnitudes and 
opposite directions. If the two objects are labelled A and B, the forces they 
exert on each other are,
on B on A
• During a collision, these forces act for the same amount of time. If the 
collision begins at time ti and ends at time X 2, then the time duration of the 
collision is
A t
, and the impulse experienced by object A is,
— » -t
h = F E o n A ^ •
• The impulse experienced by object B is,
1 b ~ F a onB&t
• If the values for the forces in these impulse equations are substituted in to 
the equation for Newton's Third Law, the result is,
^A a n B ^ B on A
. / h _ J a 
At i t
By the impulse-momentum theorem, this is equivalent to,
Pb , 2 ~ P b , 1 = ~ (P a ,2 ~ P a .l )
• In this equation,
pH
means the momentum of object A at time ti,
Pa ,2
means the momentum of object A at time X 2,
Pb, 1
means the momentum of object B at time ti, and
Pb, 2
means the momentum of object B at time X 2.
• The equation can be rearranged to put all of the terms for time ti on one side, 
and terms for time X 2 on the other,
Pa, 1 T Pb, i = Pa,2 + Pb,2
• In this case, in which there were no external forces, the sum of the momenta 
before the collision is equal to the sum of the momenta after.
• In general, as long as there are no external forces, the total momentum of the 
system is constant. This is known as conservation of momentum.
• For any number of objects the total momentum can be labeled
p
P = P a + P b + ¦ " = ™ A V a + m B V B + ¦ "
If there are no external forces, the total momentum
p
remains constant, even if the momenta of the individual objects change.
Instantaneous Impulse: Example: bat and ball contact
J = f F dt = ? LP= P f-P i
• The relation between impulse and linear momentum can be understood by the 
following equation.
Ft = m(v- u)
Where, F = Force, t = time, m = mass, v = initial velocity, u = final velocity
• Rotation about a fixed point gives the three dimensional motion of a rigid 
body attached at a fixed point.