☰ Contents · Physics

Rule of moments and dynamics of circular motion

Lessons 18–19 · 2 lessons · N. Sh. Turdiyev, K. A. Tursunmetov, A. G. Ganiyev, K. T. Suyarov, J. E. Usarov, A. K. Avliyoqulov. Physics Grade 10, 1st edition. Niso Poligraf Publishing House, Tashkent, 2017
19

Dynamics of rotational motion

Textbook: pp. 65–67
GoalCalculate the dynamics of circular motion: centripetal force, banked roads and speed in a vertical circle.
New words
centripetal force · markazga intilma kuchcentrifugal inertial force · markazdan qochma inersiya kuchispeed limit on a turn · tezlik chegarasivertical circle · vertikal aylana
Explanation

In uniform motion along a circle the net force on a body is directed along the radius toward the centre: Fc = mυ²/R = mω²R = m·4π²R/T². It is not a separate kind of force but the resultant of the actual forces: string tension, friction, and components of weight and reaction. In a rotating (non-inertial) frame an equal and opposite centrifugal inertial force Fcf = mυ²/R is introduced. On a horizontal turn friction holds the car: μmg ≥ mυ²/R, so the greatest speed is υ = √(μgR). If the road is banked inward at an angle α, the speed at which no friction is needed is υ = √(gR·tanα). In a vertical circle a body on a string or in a bucket will not fall at the top if mg ≤ mυ²/R, i.e. υ ≥ √(gR); that is why water does not spill from a swung bucket. Drivers must slow down on turns on wet roads or ice, where μ is small.

Worked examples
A car (1000 kg) takes a turn of R = 50 m at 10 m/s: Fc = 1000·100/50 = 2000 N. On a wet road with μ = 0.2 the greatest speed is √(0.2·10·50) = √100 = 10 m/s.
A bucket (R = 0.9 m) is swung in a vertical circle: for the water not to fall at the top υ ≥ √(gR) = √(10·0.9) = 3 m/s. For a turn of radius R = 160 m and speed 20 m/s, the friction-free banking angle is given by tanα = υ²/(gR) = 400/1600 = 0.25, α ≈ 14°.
Class activity

In the school yard, only under the teacher’s supervision: put a little water in a small plastic cup tied to a string and swing it slowly in a vertical circle. Feel the smallest speed at which the water does not spill. Keep everyone clear and do not swing fast.

Practice
1
What is the formula of the centripetal force and where does it point?
2
A 0.5 kg stone moves on a string in a horizontal circle of R = 1 m at 4 m/s. What is the tension (N)?
3
A turn with R = 100 m on a wet road (μ = 0.1, g = 10 m/s²). What is the greatest speed (m/s)? (υ = √(μgR))
4
Why is the outer rail raised above the inner one on railway curves?