17
Centripetal acceleration
Textbook: pp. 62–68
GoalKnow and compute centripetal acceleration a = υ²/R = ω²R.
New words
centripetal acceleration · markazga intilma tezlanishradius R · aylana radiusi Rdirection of acceleration · tezlanish yo‘nalishia = υ²/R · a = υ²/R
Explanation
In uniform circular motion the magnitude of the linear speed is constant but its direction keeps changing, so the body has an acceleration. It points toward the centre of the circle and is therefore called centripetal acceleration. Its magnitude is a = υ²/R; with υ = ωR we also get a = ω²R. The larger the speed and the smaller the radius, the larger the acceleration. The acceleration is perpendicular to the velocity vector, so it changes only the direction, not the speed. When a car turns fast, the passengers feel pushed “outward”.
Worked examples
υ = 10 m/s, R = 50 m: a = 10² : 50 = 2 m/s².
ω = 2 rad/s, R = 5 m: a = ω²R = 4 · 5 = 20 m/s².
Class activity
“Speed in a bend”: the class works out how much a grows if υ doubles in a bend (4 times). Safety: always wear a seat belt.
Practice
1
Which way does the centripetal acceleration point?
Toward the centre of the circle.
2
υ = 6 m/s, R = 9 m. a (m/s²)?
4
3
ω = 3 rad/s, R = 2 m. a (m/s²)?
18
4
Why does doubling the speed quadruple the centripetal acceleration?
Because a = υ²/R depends on the square of speed: (2υ)² = 4υ².