7.2. Centripetal Forces http://www.ck12.org
7.2 Centripetal Forces
Summary
Forces that cause objects to follow circular paths — case 3 above — are known ascentripetal, or ’center seeking’,
forces. Such forces mustcontinuously change directionto stay perpendicular to the velocity vector. We saw in the
chapters on vectors and kinematics that vectors cannot impact motion in directions perpendicular to them. This is
why the horizontal velocity of projectiles on earth does not change (as in two dimensional motion).
Speed vs. Direction
Think of a ball rolling horizontally off a cliff. At first, its velocity is perpendicular to the force of gravity. As it falls,
its velocity in the x direction stays constant, but it accelerates downward due to gravity. This ball will not travel in
circle, though, because gravity isonlyperpendicular to its velocity at the instant it leaves the cliff. Eventually, the
ball’s velocity components will be equal. After some time, if the cliff is tall enough, the ball’s vertical velocity will
dwarf its horizontal velocity.
Let’s now compare how gravity affects the ball’sspeedat different 1 second intervals during its flight.
- Let’s say this ball initially had a horizontal velocity — and therefore also speed — of 100 m/s, and a vertical
speed of 0. After the first second, its vertical velocity will about 10 m/s (assumeg=10m/s. Using the
pythagorean theorem, we find that the speed is now
√
1002 + 102 ≈ 100 .5m/s, a change of less than .5 m/s.- Now consider the ball after 10 seconds, when its velocity components are equal. Between the 10th and 11th
second, its speed goes from
√
1002 + 1002 ≈ 141 .4m/s to√
1002 +( 100 + 10 )^2 ≈ 148 .7m/s, a much bigger
increase in the same time.- Finally, for the sake of argument, let’s say the cliff is mount Everest and the ball keeps falling for 100
seconds. Now, in one second, its speed goes from
√
1002 +( 1000 )^2 ≈1005m/s to√
1002 +( 1000 + 10 )^2 ≈
1014 .9m/s.