Transcript Slide 1
Velocity versus Speed
Newton’s 2nd Law
Friction as Non-conservative Force
Velocity versus Speed
Pay particular attention when distinguishing
between the velocity vector and its scalar
magnitude, speed. Which quantity is being used is
often implied through context but a distinction
should be made. The notational differences are
Velocity is the time derivative of the position vector.
Speed is the magnitude of the velocity vector.
v(t ) v(t )
Similarly, acceleration is the time derivative of the
Newton’s 2nd law of motion states that the acceleration a of an object
of constant mass m is proportional to the vector sum of the forces F
acting on it.
F (total) ma
The 2nd law is also applied to the individual components.
Fx ,total m ax
Fy ,total m ay
Fz ,total m az
While a non-zero acceleration always implies a change in velocity, it
does not always imply a change in speed. How can this be true?
Near the surface of the earth, the force of
gravity does not vary significantly with height.
We assume the force of gravity is constant in
the cases we consider in this class.
In a Cartesian system with the j direction
pointing up, the gravitational force on mass m
F = -mgj
where g is the magnitude of gravity near the
surface of the earth, g=9.81m/s2.
The total mechanical energy of a system is the sum of the kinetic
energies (KE) of motion and potential energies (PE) associated with
the position of the system in space.
We express kinetic energy in two useful ways:
There are many sources of potential energy, e.g.
Rotational – about the center of mass of the object
Roller coasters deal with both expressions of kinetic energy but
traditionally only gravitational potential energy after the lift hill.
Kinetic Energy (KE)
Translational: KE t = ½mv2
Rotational: KE r = ½Iω2
(m=mass, v=velocity, I=moment of inertia, ω=angular velocity)
Potential Energy (PE)
Gravitational: PE = mgh
(g=gravitational acceleration near the earth’s surface, h=height)
Total Mechanical Energy (E)
E = KE t + KE r + PE = ½mv2 + ½Iω2 + mgh
Conservation of Energy (work-energy theorem)
Assuming there is no energy lost to non-conservative
forces, the total mechanical energy of the system remains
Friction is the best example of a non-conservative force –
more on friction later – but for now, assume no energy lost
Therefore, Ef = Ei where f stands for final, i for initial
½mvf2 + ½Iωf2 + mghf = ½mvi2 + ½Iωi2 + mghi
Example – assume a point mass (so neglect rotation term)
½mvf2 + mghf = ½mvi2 + mghi
How can you use this relationship if you know that the initial
velocity is zero and the final height is zero?
Friction is a non-conservative force. By doing work on the system, friction
changes the total mechanical energy.
The magnitude of the force of friction (f) is found by:
f = μN
μ = coefficient of friction, N = normal force
The direction of f is always opposite to the direction of motion
There are three types of friction
Static (μs) - no motion or sliding
Kinetic (μk) - sliding or slipping motion
Rolling (μr) - type of static friction, assuming the ball isn’t slipping as it
rolls; the inertia of the ball keeps the ball rolling so the frictional force is
due only to deformations or sticking of the two surfaces and is
significantly less than the regular coefficient of static friction
There is a difference between neglecting friction and a negligible frictional