Topic 6: Circular motion and gravitation Flashcards
Circular motion
direction of motion is constantly changing, therefore velocity is constantly changing. Acceleration is the rate of change of velocity
(circular motion) speed formula - v
v = (2πr)/T
circular acceleration formulas
a = v2/r
a = (4π2r)/T2
centripetal force formulas
F=ma
F=(mv2)/r
F=m(ωr)2/r → F=mω2r
angular velocity
the angular displacement divided by the time taken
linear velocity
v=ωr
centripetal acceleration
acceleration is towards the centre of the circle, caused by centripetal force in the same direction. e.g gravity on a satellite
How far will an object travel in one revolution?
the object will travel through 2π rads in a time T
In what way does the centripetal force act?
It always acts at 90 degrees to the motion of the object undergoing circular motion. Therefore it does not do any work - no displacement of the object in the direction of the force.
What happens to a car if the corners are banked?
if a bend is banked, a car travels much faster than on a flat road as friction is not required to produce the centripetal force - instead, it comes from the horizontal component of the reaction force.
How can you resolve the reaction force?
It can be resolved into horizontal and vertical components
How is vertical circular motion different to horizontal circular motion?
It is different to horizontal in that the force acting on the object varies throughout the motion and the speed of the object may not be constant as energy changes between Ek and Ep
What are the forces at the top of a loop?
The two forces acting on a car are the weight force (Fg) and the reaction force of the track pushing down on the car (Fr) and Fc = Fg
What is the total unbalanced force in a vertical loop?
Fg+Fr in the downwards direction - provides the centripetal force for the circular motion
The faster the car is travelling…
the larger the centripetal force (Fc is proportional to v squared).
The weight force is constant, so it is a reaction force that changes in size as the speed of the car changes (higher the speed, greater reaction force)
At the minimum speed needed to move in a vertical circle…
the reaction force at the top of the loop = 0
How do you calculate the minimum speed in a vertical circle?
Fc = Fg so (mv2)/r = mg
v2/r = g
vmin = √rg
We feel weightless at the top of a loop because…
the weight we feel actually depends on the size of the reaction force, not the size of the weight force - when reaction force is increased or decreased we feel lighter and heavier respectively. Top of loop, reaction force = 0
Why do we feel heavier at the bottom of the loop?
Fc is directly upwards towards the centre of the loop.
Using Fc=Fr-Fg, we see that as Fr > mg, therefore we feel heavier
We feel heavier in a loop when…
speed increases and/or reaction force decreases
Newton’s universal law of gravitation
every single point mass attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of their separation.
What is a point mass?
A mass which does not take up any space (infinitely small).
Gravitational force formula
F=G(Mm/r2) - both masses will experience the same size force e.g you pull the earth towards you with the same force that it pulls you towards it. The force depends on the masses involved though, and the distance between each other
What is a gravitational field?
A region of space where a mass experiences a force because of its mass (a gravitational force).
What happens to the field lines when there is a stronger field?
The field lines are closer
What is gravitational field strength?
The force, per unit mass, experienced by a small point mass placed in the field.
What is a test mass?
A mass that has such a small mass that it does not change the gravitational field in which it is placed
If a test mass of mass m experiences force F then…
g=F/m
What is g (measured in Nkg-1)
field strength; a vector quantity (has direction as well as size) and will be the same as the acceleration due to gravity at that point (ms-2 = Nkg-1) therefore a=F/m
What is the centripetal force for any orbiting object?
Centripetal force = gravitational force
What does the speed required to maintain an orbit depend on?
It depends on the radius of the orbit but not the mass of the satellite.
What is the acceleration of a satellite?
a = F/m or a=v2/r
What are geostationary orbits?
Orbits around the earth such that they stay in the same point above the earth’s surface. To do this the satellite must orbit above the equator with an orbital period equal to the time it takes the earth to rotate once.
What is the time Earth takes to rotate once in seconds?
24(hours)x60(minutes)x60(seconds) = 86400s
How do you derive Kepler’s Third law?
r3 = GMT2/4π2 where G, M, 4 and π2 are all constants so r3 is proportional to T2 and r3/T2 = constant (for any planet this ratio will be constant).
What is work done?
Force x distance moved in the direction of the force, and also a transfer of energy
What is the period (T)?
The time taken for an object to travel once around the circle (one revolution) (s)
What is frequency?
The number of times an object travels around the circle per second (Hz)
What is angular displacement?
The angle moved around the circle by an object from where its circular motion starts
Is angular displacement a vector or scalar?
Scalar (not considered a vector like linear displacement is)
What is a radian?
The angle for which the arc length is equal to the radius
How do you go from degrees to radians?
((Angle in degrees) x 2π)/360
How do you go from radians to degrees?
((Angle in radians) x 360)/2π
For what angles can sin x and x be approximated?
For angles of 10° and under or π/9 rads and lower
Why is an object in simple circular motion always accelerating?
Velocity is constantly changing, therefore it is accelerating. This means that there must be an unbalanced force.
What is the direction of angular velocity?
Through the the centre of the circular movement in the direction that the motion would be clockwise around it.
What are the units of angular speed?
Radians per second (rad s-1)
How is frequency linked to angular speed?
As angular speed is 2π/T. Frequency is 1/T. Therefore angular speed is 2πf too
Derive linear velocity with respect to angular velocity
V = d/t so 2πr/T. As angular velocity = 2π/T we can substitue this into the linear velocity so V = ωr
Which law explains the existence of centripetal acceleration.
Newton’s first law. Changing velocity indicates an external force to cause this acceleration.
In which direction does the centripetal acceleration act in?
It is always in the direction towards the centre of the circle.
If turning on a horizontal road, what condition must be met in order for the car not to skid?
The centripetal force requires has to be less that the frictional force:
mv2/r smg
When should you use the static or dynamic coefficients of friction for a car turning?
You should use the static when turning, as the frictional force has not been exceeded, if not skidding.
Once skidding, the dynamic coefficient should be used as the frictional force was exceeded and not is dynamic. The sign should also switch to an = sign.
How can you find the maximum speed of car turning on a flat road?
mv2/r = μdmg
v2/r = μdg
v2 = μdrg
vmax = +√μdrg
What is the horizontal component of the normal force (banked corners)?
N sinθ
What is the centripetal force for banked corners, in terms of weight?
Fc = N sinθ
as mg = N cosθ (= the vertical component)
N = mg/cosθ
Fc = (mg/cosθ) sinθ
Fc = mg tanθ
How do you equate the general centripetal force formula and the banked corner formula?
as Fc = mg tanθ and Fc = mv2/r
mg tanθ = mv2/r
tanθ = v2/rg
Examples of banked corners in use
- cyclist tracks (velodromes)
- aeroplanes turning
- high speed trains turning
At which point in the verticle circle is the tension most likely to break and why?
At the bottom point as the string tension must overcome weight whilst providing the required centripetal force, therefore it is most likely to be strained and break here.
What is the equation for the maximum tension a string can withstand in a verticle circle system?
For it to stay intact:
Tmax > mv2/r + mg
What is the equation for the maximum linear speed at the bottom of a verticle circle?
√(r/m)(Tmax-mg)
[derived from rearranging the maximum tension equation]
At what speed does a car going over the bridge lose contact with the road?
At a speed greater or equal to √gr
[derived from equation with car at the top of the bridge]
What is the centripetal force of a car on the top of a bridge?
It is entirely the weight
Fc = mg = mv2/r
How is energy conserved for a car driving over a bridge?
Kinetic energy at the top + change in gravitational potential energy = kinetic energy at the bottom
(1/2)mvtop2 + mg(2r) = (1/2)mvbottom2
with this you can find that:
Tbottom = Ttop + 6mg
What can the gravitational field strength also represent?
The acceleration that a mass would experience from being pulled towards another mass
