centripetal force.
(c) The force of friction provides the centripetal force here, and is equal to μ times the normal force.Because the only forces acting vertically are F (^) N and mg, F (^) N = mg . The net force is equal to mv 2
/r , and also to the friction force mmg . Setting these equal and solving for v ,
Plug in the values given (r = 0.25 m, μ = 0.30) to get v = 0.87 m/s. If the speed is faster than this,
then the centripetal force necessary to keep the penny rotating increases, and friction can no longer
provide that force.
Rapid Review
• When an object travels in a circle, its velocity vector is directed tangent to the circle, and its
acceleration vector points toward the center of the circle.
• A centripetal force keeps an object traveling in a circle. The centripetal force is simply whatever net
force is directed toward the center of the circle in which the object is traveling.
• Newton’s law of gravitation states that the gravitational force between two objects is proportional to
the mass of the first object multiplied by the mass of the second divided by the square of the distance
between them. This also means that the gravitational force felt by one object is the same as the force
felt by the second object.
• Predictions of Kepler’s laws: Planets undergo elliptical orbits with the sun at one focus; at points in an
orbit closer to the sun, a planet moves faster; the smaller a planet’s mean distance from the sun, the
shorter its orbital period, by T 2 R 3 . (The symbol “ ” means “is proportional to.”)