What kind of force is required to keep an object moving in a circle at a constant speed?
A constant centripetal force (a force always applied towards the centre of that circle)
An object moving in a circle at a constant speed is accelerating (true or false)
True
The direction is always changing hence the velocity is always changing, where acceleration is defined as the change in velocity over time
equation for velocity of uniform circular motion
v = 2pir/T
T = time for one rotation
angular velocity equation
w = linear velocity/radius = 2pi/T = 2pif = Θ/t
as f = 1/T
f = frequency of the rotation
T = object's time period = time for one rotation
w = radians per second
velocity equation
v = wr
Time for one rotation equation
T = 2pi/w
angular displacement equation
Θ = wt = 2pit/T = 2pitf
speed equation
v = 2pir/T = wr
distance equation for an object in uniforn circular motion
s =vt = 2pirt/T = Θr
angular acceleration equation
A = w^2 x r = v^2 /r
centripetal force equation
F = mv^2 / r = mw^2xr
What is a radian?
The angle of a circle sector such that the radius is equal to the arc length
Over the top of a hill
mg - S = mv^2 / r = centrpetal force
The vehicle would loose contact if its speed is equal to or geater than a particular speed, then if this happens then the support for is zero so...
mg = mv^2 / r
thus the vehicle shoud not exceed v^2 = gr
On a roundabout
The centripetal force is provided by the sideways friction force between the vehicle's tyres and the road surface
friction = centripetal force = mv^2 / r
limiting force of friction = mv^2 /r = μmg
maximum speed for no slipping v^2 0 = μgr
On a banked track
On a banked track the speed can be high if the centripetal force is weight so the speed for no sideways friction can be calculated
horizontal components = (N1 + N2)sinΘ = centripetal force = mv^2 / r
vertical components = (N1 + N2)cosΘ = weight = mg
tanΘ = (N1 + N2)sinΘ / (N1 + N2)cosΘ = mv^2 / mgr = v^2/gr
the speed for no sideways friction ...
v^2 = grtanΘ
as for no sideways friction the horizontal component for the normal force acts as the centripetal force
The big dipper
The difference between the support force and the weight acts as a centripetal force
S - mg = mv^2 / r
S = mg + mv^2 / r
So the extra force you experience due to circular motion is mv^2 / r
The very long swing
The maximum speed occurs when the swing passes through the lowest point
1/2 x mv^2 = mgh
v^2 = 2gh
The centripetal force is the difference in the support force and weight
S - mg = mv^2 / L
At maximum height
The reaction and the weight provide the centripetal force
mg + R = mv^2 / r(radius)
at a particular speed v^2 = gr then R = 0 so there would be no force on the person due to the wheel as R =mv^2 / r - mg