Chapter 16 - Circular Motion Flashcards
define the radian
“A radian is the angle subtended by a circular arc with length equal to the radius of the circle”
what is the equation for the radian
Angle (radians) = arc length/radius
define angular velocity
“the angular velocity, (lowercase omega w), of an object moving in a circular path is defined as the rate of change of angle”
what are the three equations for angular velocity
w = delta theta/ time
w = 2(Pi)/T
(T is period)
w = 2f(Pi)
(f is frequency)
what are the units for angular velocity
radS^-1
usually
why is something moving in a circular path accelerating and what does this mean
- when moving on a circular path at a constant speed your velocity is still changing because the direction of travel changes and velocity is a vector
- this means there is an acceleration
- this means there must be a resultant force
what is the name of this resultant force acting when an object moves on a circular path
the centripetal force
what are the key features of the centripetal force
- it acts perpendicular to velocity
- it acts towards the centre of the circle
- it has no ‘horizontal’ component in the same direction as the object’s tangental speed so the speed of the object remains constant
how can we derive the equations for linear velocity and angular velocity
speed = distance/time
distance in this case = 2r(pi)
and time = T (period)
because it’s the circumference of the circle so
v = 2r(pi)/T
as w = 2(pi)/T this gives us
V = wr
what is the equation linking linear and angular velocity
v = wr
what is the derivation for the equation using acceleration and radius
- imagine a small section of a circle, radius r
- the angle is deltatheta, the arc length is delta(x), the velocity at one end of the arc is Va and Vb at the other
By definition delta(theta) = delta(x)/r
As Va and Vb act in different directions there must have been an acceleration, this is given by
a = delta(v)/delta(t)
as delta(x) is small, we can model it as a straight line so delta(x) = v x delta(t)
substituting gives
delta(theta) = vdelta(t)/r
as delta(theta) also = delta(v)/v
this means
vdelta(t)/r = delta(v)/v
so v^2/r = delta(v)/delta(r) = a (from part 1)
thus V^2/r = a
what are the two equations for acceleration on a circle
a = V^2/r
and
a = w^2 x r
how to derive the two centripetal force equations
- we know F = Ma
- we know a = V^2/r and a = (w^2)r
so therefore
F = mv^2/r
F = m(w^2)r
what are the three derivations you need to know
1) derivation of v = wr
2) derivation of a = V^2/r and a = w^2r
3) derivation of F = mv^2/r and F = mw^2r
what is an experiment we can do to investigate circular motion
- set up equipment where there is a mass on a string, passing through a metal/glass tube, attached to another mass
- the length of the string above the tube is r
- we know that the faster we swing the mass, the more likely the hanging mass is to rise up
- this is because swinging the mass faster means it requires a greater centripetal force, if the weight of the other mass isn’t sufficient then the weight will rise up due to tension in the string
- if the centripetal force required is lower than the weight of the hanging mass then the hanging mass will drop
- this along with our equations allows us to do a variety of experiments and plot different graphs
what is a centrifuge and briefly explain how it works
- a centrifuge is a device which spins samples of liquids in tubes in order to separate the components of the liquid
- the heavier components of the liquid will travel to the outside (bottom of the tube) because they will require a greater centripetal force to keep them in place
- the lighter components of the liquid will remain on top
why do banked surfaces effectively increase the centripetal force
- a component of the normal contact force acts horizontally towards the centre
how do banked surfaces allow objects/riders to corner faster
- they increase the effective centripetal force acting on the object
- this means they can travel faster before the required centripetal force is too high
how can normal contact forces be changed due to centripetal forces
- where there is a centripetal force, the normal contact force acting on an object will be lower/higher depending on if it is acting in the same direction as the weight or the opposite direction to the weight
- e.g. the centripetal force acting on an object at the equator will decrease it’s mass as it acts ‘up’ on the object
- but a centripetal force acting on an object at the bottom of a loop will increase the effective normal contact force
what is a conical pendulum
- this is a pendulum which swings around in a circle rather than back and forth
- it used to be used in smooth timing mechanisms
derive the suitable equations for the conical pendulum to show velocity and angle are independent of mass
f = ma = mv^2/r
F = Ftsin(theta)
where Ft is the tension in the string and theta is the angle between the string and the centre line
thus Mv^2/r = Ftsin(theta)
we also know that the weight of the pendulum bob is being conteracted
so Ftcos(theta) = mg
divide first by second and you get
tan(theta) = v^2/rg
For an object moving on a banked surface at angle theta, what is the equation for the vertical component of it’s normal reaction force and what is this equivalent to
N(vertical) = Ncos(theta) = W = mg
what is the easiest way to calculate the centripetal force for the bung experiment
W = mg = Fc
what to remember on questions about planes flying at angles
- the centripetal force is the HORIZONTAL component of lift
- the weight of the plane is usually the VERTICAL component of lift
- trig will likely be needed to calculate these
