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Penn State Schuylkill

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Ex 1) Circular motion horizontal surface w/ friction

A car with mass M moves in a circle of radius R on a asphalt road (µ) with a speed V.

Find V

F_c = mv²/r = µmg

v²/r = µg

v = (µgr)^-1

Ex 2) Mass on a String - Vertical Circle

Mass M attached to a string moves at constant speed of v in a vertical circle of radius r.

Find string tension T_top and T_bottom

Top

Free Body Diagram

Make center of circle +

F_net = ma

F_net = Mg + T_top

Mv²/r = Mg + T_top

M[v²/r - g] =  T_top

F_net = ma

F_net = T_bottom - Mg

Mv²/r = T_bottom - Mg

M[v²/r + g] = T_bottom

Mv²/r = Mg + T_top

Mv²/r = T_bottom - Mg

Since v is constant ...

Mg + T_top = T_bottom - Mg

2Mg + T_top = T_bottom

Hammer Throw

Ex 3) Object mass M moves at constant speed V at the end of string in a circle r.

What is the minimum speed needed to maintain the object's circular motion?

What is T_top at minimum speed?

T_top = 0

Free Body Diagram

Make center of circle +

F_net = ma

F_net = Mg + T_top

Mv²/r = Mg

When T_top = 0

v²/r = g

Ex 4) Rollercoaster - velocity changes with height

Car of mass M, circle radius r,
speed v

Find Normal Force at top

Freebody diagram at top:

F_net = mv²/r

F_n + mg = mv²/r

F_n = [mv²/r] - mg

What minimum velocity is required for the roller coaster to prevent the passengers from falling off?

 Minimum v occurs when F_n = 0

F_n = [mv²/r] - mg

 [mv²/r] = mg

v²/r = g

Fc Banked Track

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