1.1 - Kinematics
Trajectories
• A point mass is an abstraction. It can be used to describe the position of an
object when its spatial extent is irrelevant.
• The position of a point mass can be uniquely determined after defining a
coordinate system. This position can be described by using the position vector r
= {x,y,z}.
- Origin (r={0,0,0}
- Scale/unit (how far is x=1 from x=0)
- Orientation (direction in which the axes point)
• When the position of a point mass changes with time, the function
r(t)={x(t),y(t),z(t)} describes its trajectory
Position and velocity
• Displacement between two points: difference between the two position vectors
(Δr = r1 - r0), this is independed of the trajectory followed.
• The uniform linear motion of a point mass can be written as r(t) = v * t with v =
{vx, vy, vz} = const
• Over a time interval the PM moves by Δr = v * Δt, on a straight line and with the
same magnitude (linear and uniform motion)
• In an (r,t)-graph the velocity represents the slope of the position vs the time
curve:
- Average velocity: v = Δr(t)/Δt
- Instantaneous velocity: v(t) = dr(t)/dt = r'(t)
- Magnitude of the velocity: |v|
Acceleration
• If velocity changes with time, we can define acceleration a(t) as the slope of the
velocity vs time curve:
- Average acceleration: a = Δv(t)/Δt
- Instantaneous acceleration: a(t) = v'(t) = r''(t)
• Uniformly accelerated motion: acceleration vector is constant.
- Velocity can be obtained by integration: v(t) = a * t + b
- Position equation: r(t) = 0.5at2 + v0t + r0
- In one dimension, the equations for velocity and position read
v(t) = v0 + at
x(t) = x0 + v0t + 0.5at2
Projectile motion
• If, in the presence of (constant) gravity, a body is given an initial velocity, it will
follow a parabolic trajectory.
- x(t) = v0xt
z(t) = -0.5gt2 + v0zt + h
Uniform circular motion
• Motion with constant velocity along a circular path is a non-constant
acceleration that only changes direction. The component along the path is zero,
the acceleration vector is a centripetal force.
• Magnitude of centripetal acceleration: ac = v2/R = 2 * R
- = v/R
Mechanics - ECTBY1Q1 Page 1
,1.2a - Forces
Forces as vectors
• An object can only change motion if acted upon by a force.
• A force is a vector. If several forces act on a body, the net force is the vector sum of the
individual forces (superposition principle)
Newtons laws
• First law of Newton
- Momentum, the state of motion of a particle m: p = m*v.
- A particle on which no force is acting is called a free particle.
- First law of Newton: the motion of a free particle is constant in time: p(t) = const.
• Second law on Newton
- We define force as the change in momentum over time: F = dp/dt
- If mass is constant over time: F = m * a
• Third law of Newton
- When two bodies interact with each other, the forces they exert on each other are equal
in magnitude and opposite in direction: F12 = -F21
Inertial and gravitational mass
• The mass in Newton's equations is an intrinsic property of a body, it can also be referred to as
inertial mass.
• Inertial and gravitational mass are the same.
Mechanics - ECTBY1Q1 Page 2
, 1.2b - Energy, work & friction
Work
• If a point mass moves over a distance ∆ when subject to a constant force F, we
define the mechanical work on the point mass due to F as the scalar product.
W=F*∆
• If the point mass follows a non-linear path or the force is not constant, the work
by F along the path is: W = F(r)*dr
• If force is perpendicular to the displacement, the work is zero.
• Work done over a unit of time is called power: P = dW/dt = F*v
Conservative forces
• Force fields are conservative when the work is independent from the followed
path.
• Force fields that are homogeneous or central are always conservative.
• Spring force is also conservative.
Potential energy
• Since the work done by a conservative force field is only dependent on initial and
final position, it is possible to define a function of position called potential
energy, such that W = -∆Ep
• W = ∆Ekin where kinetic energy is defined as Ekin= 0.5mv2
• Net work: work done by the net force.
Law of conservation of mechanical energy
• In the special case when only conservative forces are present, the work done on
an
object is also equal (and opposite) to the change in its potential energy.
• Combining the definitions of potential and kinetic energy, we obtain: E = ∆Ep +
∆Ekin Where E is the total mechanical energy of the body.
Static friction
• When two objects are in contact with no relative motion, they can only start
sliding along
each other if the force along the contact surface is larger than the force of static
friction S =
S ∙ N , where S is the coefficient of static friction and N the magnitude of
the normal
force.
• The magnitude of the force of friction does not depend on the size of the contact
area.
• The direction of the force of friction is always opposite to the direction of the
force trying
to cause the motion.
Kinetic (sliding) friction
• When two objects are sliding along each other, the value of the force of friction is
still proportional to the normal force according to k = k ∙ N, where k is the
coefficient of kinetic friction and where k< s
• Note that friction is a non-conservative force and the work it does depends on
the path
followed by the moving object
Mechanics - ECTBY1Q1 Page 3
Trajectories
• A point mass is an abstraction. It can be used to describe the position of an
object when its spatial extent is irrelevant.
• The position of a point mass can be uniquely determined after defining a
coordinate system. This position can be described by using the position vector r
= {x,y,z}.
- Origin (r={0,0,0}
- Scale/unit (how far is x=1 from x=0)
- Orientation (direction in which the axes point)
• When the position of a point mass changes with time, the function
r(t)={x(t),y(t),z(t)} describes its trajectory
Position and velocity
• Displacement between two points: difference between the two position vectors
(Δr = r1 - r0), this is independed of the trajectory followed.
• The uniform linear motion of a point mass can be written as r(t) = v * t with v =
{vx, vy, vz} = const
• Over a time interval the PM moves by Δr = v * Δt, on a straight line and with the
same magnitude (linear and uniform motion)
• In an (r,t)-graph the velocity represents the slope of the position vs the time
curve:
- Average velocity: v = Δr(t)/Δt
- Instantaneous velocity: v(t) = dr(t)/dt = r'(t)
- Magnitude of the velocity: |v|
Acceleration
• If velocity changes with time, we can define acceleration a(t) as the slope of the
velocity vs time curve:
- Average acceleration: a = Δv(t)/Δt
- Instantaneous acceleration: a(t) = v'(t) = r''(t)
• Uniformly accelerated motion: acceleration vector is constant.
- Velocity can be obtained by integration: v(t) = a * t + b
- Position equation: r(t) = 0.5at2 + v0t + r0
- In one dimension, the equations for velocity and position read
v(t) = v0 + at
x(t) = x0 + v0t + 0.5at2
Projectile motion
• If, in the presence of (constant) gravity, a body is given an initial velocity, it will
follow a parabolic trajectory.
- x(t) = v0xt
z(t) = -0.5gt2 + v0zt + h
Uniform circular motion
• Motion with constant velocity along a circular path is a non-constant
acceleration that only changes direction. The component along the path is zero,
the acceleration vector is a centripetal force.
• Magnitude of centripetal acceleration: ac = v2/R = 2 * R
- = v/R
Mechanics - ECTBY1Q1 Page 1
,1.2a - Forces
Forces as vectors
• An object can only change motion if acted upon by a force.
• A force is a vector. If several forces act on a body, the net force is the vector sum of the
individual forces (superposition principle)
Newtons laws
• First law of Newton
- Momentum, the state of motion of a particle m: p = m*v.
- A particle on which no force is acting is called a free particle.
- First law of Newton: the motion of a free particle is constant in time: p(t) = const.
• Second law on Newton
- We define force as the change in momentum over time: F = dp/dt
- If mass is constant over time: F = m * a
• Third law of Newton
- When two bodies interact with each other, the forces they exert on each other are equal
in magnitude and opposite in direction: F12 = -F21
Inertial and gravitational mass
• The mass in Newton's equations is an intrinsic property of a body, it can also be referred to as
inertial mass.
• Inertial and gravitational mass are the same.
Mechanics - ECTBY1Q1 Page 2
, 1.2b - Energy, work & friction
Work
• If a point mass moves over a distance ∆ when subject to a constant force F, we
define the mechanical work on the point mass due to F as the scalar product.
W=F*∆
• If the point mass follows a non-linear path or the force is not constant, the work
by F along the path is: W = F(r)*dr
• If force is perpendicular to the displacement, the work is zero.
• Work done over a unit of time is called power: P = dW/dt = F*v
Conservative forces
• Force fields are conservative when the work is independent from the followed
path.
• Force fields that are homogeneous or central are always conservative.
• Spring force is also conservative.
Potential energy
• Since the work done by a conservative force field is only dependent on initial and
final position, it is possible to define a function of position called potential
energy, such that W = -∆Ep
• W = ∆Ekin where kinetic energy is defined as Ekin= 0.5mv2
• Net work: work done by the net force.
Law of conservation of mechanical energy
• In the special case when only conservative forces are present, the work done on
an
object is also equal (and opposite) to the change in its potential energy.
• Combining the definitions of potential and kinetic energy, we obtain: E = ∆Ep +
∆Ekin Where E is the total mechanical energy of the body.
Static friction
• When two objects are in contact with no relative motion, they can only start
sliding along
each other if the force along the contact surface is larger than the force of static
friction S =
S ∙ N , where S is the coefficient of static friction and N the magnitude of
the normal
force.
• The magnitude of the force of friction does not depend on the size of the contact
area.
• The direction of the force of friction is always opposite to the direction of the
force trying
to cause the motion.
Kinetic (sliding) friction
• When two objects are sliding along each other, the value of the force of friction is
still proportional to the normal force according to k = k ∙ N, where k is the
coefficient of kinetic friction and where k< s
• Note that friction is a non-conservative force and the work it does depends on
the path
followed by the moving object
Mechanics - ECTBY1Q1 Page 3
