Download Conservation of linear momentum

Unit 4-Physics On The Move
Topic 1-Further Mechanics
Use the expression p=mv
Momentum = mass x velocity: i.e. p=mv, units kg m s⁻¹
Momentum is a vector quantity, with both magnitude and direction.
Investigate and apply the principle of conservation of linear
momentum to problems in one dimension
Conservation of linear momentum
The linear momentum of any closed system is conserved. Linear momentum is a vector quantity and
each component is conserved independently.
When two things interact, the force exerted on one is always equal and opposite to the force
exerted on the other (Newton’s third law). The interaction time is the same for both so they receive
equal and opposite impulses. This means that impulses on two colliding objects have the same
magnitude but opposite directions, meaning that the change in momentum of the two bodies is also
equal in magnitude and opposite in direction. There is no change in the total linear momentum of
the system of the two objects. This is an example of a conservation law-the two cars have interacted
but the total linear momentum is the same before and after the collision. All interactions conserve
linear momentum, and all forces arise through interactions. So the total linear momentum of the
universe is constant. For practical purposes we apply the law in a more restricted way to closed
systems.
Newton’s third law of motion




Forces never arise singly but always in pairs as the result of interactions
When A interacts with B the force A exerts on B is always equal to the force B exerts on A but
in the opposite direction along the same line of action
Because these pairs of forces arise from an interaction they are always of the same type: for
example, both gravitational or both electromagnetic
Action-reaction pairs always act on different bodies, never on the same body
Investigate and relate net force to rate of change in momentum in
situations where mass is constant (Newton’s second law of motion)
The second law of motion
The resultant force F exerted on a body is directly proportional to the rate of change of linear
momentum p of that body
Resultant force (N) is proportional to the rate of change of linear momentum (kg m s⁻²), which is the
same as saying the resultant force is proportional to the final linear momentum minus the initial
linear momentum all divided by the time for which the force acts. Therefore, strictly speaking, F in
the equation is the average resultant force during time t. Momentum, force and velocity are all
vectors, so the momentum changes are parallel to the resultant applied forces.
∑𝐹 =
∆𝑝
∆𝑡
This statement of Newton’s second law encompasses situations in which the mass is indeterminate
(e.g. interactions involving neutrinos) or continually changing (e.g. a stream of water striking a
surface).
Impulse and momentum change
Ft=mv-mu
The quantity on the left-hand side is called the impulse, the product of force and time, and is
measured in Ns. The quantity on the right-hand side is the change of momentum produced by this
impulse. A large force acting for a short time causes the same change of momentum as a small force
acting for a longer time. You use this every time you jump off something and bend your knees on
landing. If you were to land rigidly, your momentum would fall to zero in a very short time. The large
rate of change of momentum this involves would exert large forces on your body and probably cause
injuries. Bending your knees allows the same change of momentum to occur over a longer time, and
so reduces the force on your body. A similar principle applies in the use of air bags and crumple
zones to make cars safer.
Force-time graphs
The term impulse is usually used when a force acts for a short time, but the equation works for
constant forces acting for a longer period too. If the force changes with time then total impulse is
equal to the area under a force-time graph.
Derive and use the expression 𝑬𝒌 =
𝒑²
𝟐𝒎
for the kinetic energy of a non-
relativistic particle
p=mv => p²=m²v²
So
𝑝² 𝑚²𝑣²
=
=½𝑚𝑣²
2𝑚 2𝑚
which is the kinetic energy of a particle.
Analyse and interpret data to calculate the momentum of (nonrelativistic) particles and apply the conservation of linear momentum
to problems in one and two dimensions
Collisions in two dimensions
Many collisions and interactions occur in more than one dimension. Linear momentum is a vector
quantity, and so the vector sum of all momenta after a collision or interaction must equal the
original momentum vector. To solve problems in more than one dimension, it is best to resolve all
momenta along a carefully chosen set of axes and then conserve each component of momentum
independently.
Conservation of angular momentum
The angular momentum about a particular axis is constant if there is no resultant torque acting
about that axis. Comets and planets conserve angular momentum about the Sun, since the
gravitational force acts towards the Sun and therefore generates no resultant torque about it.
Explain and apply the principle of conservation of energy, and
determine whether a collision is elastic or inelastic
Energy cannot be created or destroyed, although it can be transferred from one form to another.
This principle, and the conservation of momentum, together form the cornerstone of physics at all
levels. In collisions, both conservation principles must apply, but while the momentum of the system
after the interaction is always exactly the same as it was before the interaction, the energy can be
changed into other forms. Collisions are usually classified in terms of the kinetic energy in the system
before and after the interaction
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
In elastic collisions, all of the kinetic energy in the system is conserved
In inelastic collisions, some or all of the kinetic energy is transferred to other forms of
energy. In practice, all large-scale collisions are inelastic
It is also possible for there to be an increase in kinetic energy in a system-for example, in explosions
where some chemical energy is converted to kinetic energy of the exploded fragments, and in
nuclear radiation and nuclear fission where a mass defect is transferred as extra kinetic energy.
Rotational kinetic energy
Rotating masses can store very large amounts of kinetic energy. If something is rolling, it will have
both translational kinetic energy (of its centre of mass) and rotational kinetic energy (about its
centre of mass). The RKE is equal to the sum of the kinetic energies of all the particles in the body
because of their tangential velocities about the axis.
Express angular displacement in radians and in degrees, and convert
between those units
Rotation
2𝜋

Period of rotation (T)-The time (s) to complete one revolution about some axis. T= 𝜔

Frequency (f)-The number of revolutions per second (Hz). f= =
1 𝜔
𝑇 2𝜋

Angular displacement (θ)-The angle (in radians) turned through some time. Θ=
𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ
.
𝑟𝑎𝑑𝑖𝑢𝑠
2π
radians = 360°

Angular velocity (ω)-The rate of change of angular displacement (rad s⁻¹). 𝜔 =
∆𝜃
∆𝑡
𝑣
=𝑟
Explain the concept of angular velocity, and recognise and use the
relationships v=ωr and T=
𝟐𝝅
𝝎
Angular velocity and tangential velocity
When something moves in a circle, its instantaneous linear velocity is always parallel to a tangent to
the circle; this is called its tangential velocity. It has no radial velocity. If it moved along a spiral path
it would have both a tangential and a radial velocity at all times. Angular velocity is related to
tangential velocity.
Explain that a resultant force (centripetal force)is required to
produce and maintain circular motion
Uniform circular motion
When an object moves in a circle, the force which changes the direction of the moving object is
directed in towards the centre of rotation. This is called a centripetal force. If a centripetal force is
removed, the object concerned is free to continue moving in a straight line. A centripetal force acts
at right angles to velocity, so it has no component parallel to motion. It therefore leaves the
tangential velocity and kinetic energy unchanged. It does no work on the object; it merely changes
the direction of its velocity vector. In circular motion problems you should be aware of the cause of
the centripetal force. Examples include the tension in a string when a mass attached to the end of
the string is whirled in a circle, and the frictional force of the road pushing on the tyres as a car
rounds a bend.
Resistance to twist
The moment of inertia of an object is its ‘reluctance’ to change its state of rotational motion. It is the
rotational equivalent of mass in linear mechanics. Unlike mass it depends on both the body and the
axis about which it is rotated, so the same thing has an infinite number of moments of inertia. The
moment of inertia depends on the distribution of mass about the axis of rotation.
Centripetal force and acceleration
Resultant forces cause acceleration. It follows that objects moving in circular motion are accelerating
in the direction of the centripetal force, that is, towards the centre of rotation. The magnitude of the
acceleration and force can be calculated from the rate of change of the velocity vector. Since the
magnitude of the velocity of the object remains unchanged, the centripetal acceleration must always
be directed at right angles to the velocity, towards the centre of the circle.
𝒎𝒗²
Use the expression for centripetal force F=ma=
𝒓
and hence derive
𝒗²
and use the expressions for centripetal acceleration, a= and a=rω²
𝒓