Drag the mass sliders to move each object. The centre of mass (★) repositions instantly. Try making one mass huge!
Mass 1 (m₁) 3 kg
Mass 2 (m₂) 3 kg
Mass 3 (m₃) 3 kg
Mass 4 (m₄) 3 kg
x̄ = Σmᵢxᵢ/Σmᵢ | ȳ = Σmᵢyᵢ/Σmᵢ → CoM = (—, —)
Total mass
—
kg
CoM x̄
—
m
CoM ȳ
—
m
Σmᵢxᵢ
—
kg·m
Σmᵢyᵢ
—
kg·m
CoM x̄ as each mass changes
Mass contribution to CoM
Key insight: The centre of mass is the point where the entire mass of a system can be considered to act. If m₁ ≫ m₂, the CoM is almost at m₁. For equal masses it's exactly in the middle. It's why rockets must fire through the CoM to avoid rotation!
Centripetal Force — F = mv²/r = mω²r
Centripetal force always points INWARD toward the centre. Change speed and radius — watch the force arrow scale.
Speed v 10 m/s
Radius r 5 m
Mass m 2 kg
F_c = mv²/r = — N | ω = v/r = — rad/s
Centripetal F
—
N
Angular vel ω
—
rad/s
Period T
—
s
Accel a=v²/r
—
m/s²
Freq f=ω/2π
—
Hz
F vs speed v (at current r, m)
F vs radius r (at current v, m)
Real examples: A car turning a corner — friction provides centripetal force. A satellite in orbit — gravity IS the centripetal force. A string swinging a ball — tension provides centripetal force. Without it the object flies off in a straight line (Newton's 1st Law).
Centrifugal Effect — The Pseudo-Force
In a rotating frame, objects appear to be pushed outward. This is NOT a real force — it's an inertial effect. See both reference frames side by side.
Angular speed ω 2 rad/s
Radius r 4 m
Mass m 2 kg
F_centrifugal = mω²r = — N (outward, in rotating frame only)
Centripetal F
—
N ← inward
Centrifugal F
—
N → outward
ω (angular)
—
rad/s
mω²r
—
N
Centrifugal force vs ω
Centrifugal force vs radius
Inertial vs rotating frame: In the ground frame (inertial), there is only centripetal force inward. In the rotating frame (non-inertial), we ADD a fictitious centrifugal force outward to make Newton's 2nd law work. Same physics, different perspective. Your feeling of being pushed out in a turning car is this effect!
Banking of Roads — tan θ = v²/rg
Banked roads let cars turn without relying on friction alone. The normal force provides centripetal force. Find the optimal banking angle for any speed and radius!
Speed v 20 m/s
Radius r 50 m
Friction μ 0.5
Mass m 1000 kg
tan θ = v²/rg → θ_opt = —° | v_max = — m/s
Optimal angle θ
—
°
Normal force N
—
N
Required F_c
—
N
Max speed (with μ)
—
m/s
Min speed
—
m/s
Optimal angle vs speed
Max speed vs banking angle
Why bank roads? On a flat road, only friction provides centripetal force — limited by μ. On a banked road, the normal force has a horizontal component pointing inward, providing centripetal force even without friction. Formula tracks (like velodrome or NASCAR) use very steep banking (up to 33°!) to allow very high speeds.