Horizontal Circular Motion

Car rounding a flat-curve

Car on a Flat-Curve

Question:

What is the centripetal force for a car moving along a curved road?

Answer:

Static frictional force between the road and the tires.

Centripetal acceleration

F = mamv

r

2

Smaller radius: larger force required to keep it in uniform circular motion.

Radius of Circular Path

Car travels at a constant speed around two curves. Where is the car most likely to skid? Why?

Skidding

Skidding Example

The Normal Force Can Yield a Centripetal Acceleration

How many forces areacting on the car (assumingno friction)?

Engineers have learned to “bank” curves so that cars can safely travel around the curve without relying on friction at all to supply the centripetal acceleration.

Banked Curves

Banked CurvesQ: Why the curved roads in highways are banked?

A: The normal reaction supplement the frictional force for turning.

Banked Curves - Equations

rg

v tan

2

Vertical Equilibrium : FN cos = mg

r

vFN

2

sin Horizontal Acceleration : gr

v tan

2

Banking the curve can help keep cars from skidding.

In fact, for every banked curve, there is one speed where the entire centripetal force is supplied by the horizontal component of the normal force, and no friction is required.

This occurs when:

Ideal Banking

Turning – Aeroplane

• Airplanes don’t have “rubber on the road”, so no friction to keep them from going sideways around turns

• Wings produce lift force, so proper bank angle supplies necessary horizontal component of force to produce turn

Turning - Aeroplane

Answer: To generate the centripetal force required for the circular motion.

Question: Why do airplanes make banked turn?

Airplanes in high-g turn

gravity

enhanced liftduring turn: vertical component cancelsgravity to produce level flight,horizontal componentaffects turn

Pilot accelerated by orange (lift) vector, feels heavier than normal.In this case, pilot feels about 3 g’s (orange arrow about 3 times longer than gravity arrow)

Conical Pendulum

A: The horizontal component of the tension in the string.

What force produces the centripetal acceleration?

Old-Fashioned Swings• The angle of the ropes

tells us where the forces are:

• Ropes and gravity pull on swingers

• If no vertical motions (level swing), vertical forces cancel

• Only thing left is horizontal component pointing toward center: centripetal force

• Centripetal force is just mv2/r (F = ma; a = v2/r)

gravity (mg)

swing ropes: what you feel from your seat

resultant: centripetal