Intuition The one core ideaTo move in a circle, something must constantly pull an object toward the center — and the exact pull needed is r m v 2 gravity sometimes helps and sometimes fights the pull, so the string tension must adjust — and where tension would have to become negative, the object leaves the circle.
This page builds every letter, arrow, and idea the parent note uses, starting from nothing. If you have never seen r m v 2 g r
Before any physics, we need the shape the object moves on.
r r distance from the center of the circle to the moving object , measured in metres. It never changes while the object stays on the circle — that constant distance is what makes the path a circle .
Picture a ball on a string. Your hand is the center . The taut string is the radius r
Look at the figure: the amber dot is the center, the cyan line is r
Mnemonic Always ask first"Where is the center right now?" Everything — the direction of the pull, which way gravity points relative to it — depends on the answer. At the top the center is below ; at the bottom it is above .
m m how much stuff is in the object (kilograms). More mass = harder to speed up, slow down, or turn.
Definition Gravitational fieldg g ≈ 9.8 m/s 2 how strongly Earth pulls each kilogram downward . It is a fixed number near Earth's surface, and it always points straight down , no matter where the object is on the loop.
m g force of gravity on the object is m g m g down , present at every single point of the circle.
m g neither toward nor away but sideways. The number m g role changes with position — that shifting role is the heart of the topic.
Because gravity's role changes around the loop, we need a clean way to ask "how much of this arrow points toward the center?" That question is answered by a component .
Definition Component of a forceA component of a force in some direction is how much of that force effectively acts along that direction . Picture the force's arrow casting a shadow onto a chosen line: the length of the shadow is the component. A force straight along the line has full component; a force perpendicular to the line has zero component (its shadow is a single point).
In the figure, the object sits at a general angle θ m g
a radial part m g cos θ
a tangential part m g sin θ
Intuition Why split at all?Only the radial shadow feeds the centripetal bill r m v 2 top (θ = 0 cos 0 = 1 all of m g sides (θ = 9 0 ∘ cos 9 0 ∘ = 0 nothing to turning there.
a velocity is changing per second== (units m/s²). Crucially, velocity includes direction : even at constant speed, if your direction changes, you are accelerating. Turning is acceleration.
Now we derive , not assert, the inward acceleration.
Intuition Plain-words takeawayFaster means both a quicker turn and a bigger velocity to redirect — two effects multiply, giving v × v Centripetal force and acceleration , but the picture above is the whole reason.
Now multiply the inward acceleration by mass, using F = ma
Definition Centripetal requirementTo keep an object of mass m v r total force pointing toward the center must equal exactly
m ⋅ a = r m v 2 . center-seeking . This is not a new force — it is a bill that real forces must pay .
In the figure, the object is at the top. The cyan arrow labelled r m v 2 required inward pull — it points from the object toward the center (downward here). It is drawn dashed to remind you: it is a demand , not a physical arrow you add.
Common mistake "Centripetal force is an extra arrow I draw."Why it feels right: it has a name and a formula, so it looks like a force.
The fix: draw only the real forces (gravity, tension). Then add up their radial components and set that inward total equal to r m v 2 answer forces must produce, never a separate arrow. See Free-body diagrams .
The string (or the track) supplies the rest of the inward force that gravity doesn't cover.
T T inward toward the object's anchor==. A string can pull but never push , so T ≥ 0 Tension and constraint forces .)
N normal force N push , so N ≥ 0 Normal force in circular tracks .
≥ 0 floor on how little inward help it can give: zero. If the circle's demand r m v 2 below what gravity alone provides, the string would have to push — impossible. That impossible moment defines the minimum speed . Everything in the parent note flows from T ≥ 0
This figure shows the object at the top with both real arrows: white m g T m g T T + m g = r m v 2 . T = 0 v top,min = g r
The answers come out as g r 5 g r
v 2 squared ). But we want v the operation that undoes squaring : v 2 = v v 2 = g r v = g r
Worked example Quick sanity checkv 2 = 9.8 × 1.0 = 9.8 v = 9.8 ≈ 3.13 m/s 3.1 3 2 ≈ 9.8
Definition Net-force form of Newton's Second LawF net = ma sum of all real forces on the object equals mass times its acceleration a in the radial (inward) direction , the sum of the radial components of the real forces equals m ⋅ r v 2 = r m v 2 Newton's Second Law — net force form .
This is the machine that turns "arrows on a diagram" into an equation. At every point of the loop we (1) draw real forces, (2) add their inward components (§3), (3) set the sum equal to r m v 2
The bottom-speed result 5 g r
Definition Gravitational potential energym g h Energy stored in height . h from its motion, so it slows down. See Conservation of mechanical energy .
Definition The height gain2 r bottom of the circle to the top , the object rises by two radii, h = 2 r 2 r
Intuition Why tension does no work hereTension always points along the string (radius), while motion is along the circle (tangential, §3). A force perpendicular to motion transfers no energy . That is precisely why we may use pure energy conservation between bottom and top — tension quietly drops out.
The diagram below is not decoration: follow the arrows and you get the build order of this page . Geometry and forces (top row) combine into the centripetal requirement; Newton's law turns that into the top equation; the T = 0
Circle center and radius r
Centripetal requirement mv squared over r
Split mg into radial and tangential parts
Acceleration equals v squared over r derived from turning
Newton second law radial direction
Tension T and normal force N cannot push backward
General angle equation T plus mg cos theta equals mv squared over r
Top case T equals zero gives sqrt of gr
Kinetic energy and potential energy mgh
Test yourself — you should be able to answer each before tackling the parent note.
What does r The fixed distance from center to object; it stays constant.
Which direction does m g Straight down, at every point of the loop.
What is a "component" of a force? How much of the force acts along a chosen direction — the shadow the arrow casts on that line.
What are the radial and tangential directions? Radial = along the center line (inward); tangential = along the circle, perpendicular to radial.
What is gravity's radial component at angle θ m g cos θ m g − m g
Why is turning a form of acceleration even at constant speed? Velocity has direction; changing direction changes velocity, which is acceleration.
Where does the v 2 v 2 / r One v v
Is r m v 2 No — it is the required inward total that real forces must sum to.
What sign restriction does a string obey? T ≥ 0
What is the general-angle inward equation? T + m g cos θ = r m v 2
What defines the minimum speed at the top? The moment T = 0 r m v 2
How does 5 g r v bot 2 = v top 2 + 4 g r = g r + 4 g r = 5 g r
Why can we use energy conservation with the string attached? Tension is perpendicular to motion, so it does zero work.
