1. CENTRIPETAL FORCE
When a particle moves over a curved path which is known, the equation of motion for the particle may
be written in the tangential and normal directions.
Tangential Force:
Normal Force (Centripetal Force)
Recall that tangential acceleration, at = dv/dt represents the time rate of change in the magnitude of
velocity. Consequently, if ΣFt acts in the direction of motion, the particle’s speed will increase, whereas if
it acts in the opposite direction, the particle will slow down.
Likewise, normal acceleration, an = v2r represents the time rate of change in the velocity’s direction.
Since this vector always acts positive towards the path’s center of curvature, the ΣFn, which causes an,
also acts in this direct. Since this force is always directed toward the center of the path, it is often
referred to as the centripetal force.
Magnitude of Centripetal Force
From the second law of motion, we can see that the centripetal force Fn on an object mass m in uniform
circular motion is Fn = m an which has a magnitude

2. Example:
a. Find the centripetal force needed by a 1,200 kg car to make a turn of radius 40m at a speed of
25 km/h (16mi/h).
b. Assuming the road is level, find the minimum coefficient of static friction between the car’s tires
and the road that will permit the turn to be made.
High speed turn on a level road can be dangerous. In this example, increasing the speed to 60
km/h (37mi/h) increases the needed µS to 0.72, which is too much for wet road, and the car
would skid.

3. Banking Angle
Usually the friction between the tires and the road is not enough to provide a car with the centripetal
force it needs to make a turn. If the car’s speed is high or the road surface is slippery, however, the
available force may not be enough and the car will skid. To reduce skids, the highway curves are often
banked so that the roadbed tilts inward.
The tangent of the proper banking angle θ varies directly with the square of the car’s speed and
inversely with the radius of the curve. The mass of the car does not matter. When a car goes around a
curve at precisely the design speed, the reaction force of the road provides the centripetal force. If the
car goes more slowly than this, friction tends to keep it from sliding down the inclined roadway; if the
car goes faster, friction tend to keep it from skidding outward.