Chapter 5 Uniform Circular Motion and Gravitation
5.3 Centripetal Force
Summary
- Calculate friction on a car tire moving in a circle
Any force or combination of forces can cause a centripetal or radial acceleration. Just a few examples are the tension in the rope on a tether ball, the force of Earth’s gravity on the Moon, friction between roller skates and a rink floor, a banked roadway’s force on a car, and forces on the tube of a spinning centrifuge.
Any net force causing uniform circular motion is called a centripetal force. The direction of a centripetal force is toward the center of curvature, the same as the direction of centripetal acceleration. According to Newton’s second law of motion, net force is mass times acceleration: net F = ma. For uniform circular motion, the acceleration is the centripetal acceleration— a = ac. Thus, the magnitude of centripetal force Fc is
You may use whichever expression for centripetal force is more convenient. Centripetal force Fc is always perpendicular to the path and pointing to the centre of curvature, because ac is perpendicular to the velocity and pointing to the centre of curvature.
Note that if you solve the first expression for r, you get
This implies that for a given mass and velocity, a large centripetal force causes a small radius of curvature—that is, a tight curve.
Example 1: What Coefficient of Friction Do Care Tires Need on a Flat Curve?
(a) Calculate the centripetal force exerted on a 900 kg car that negotiates a 500 m radius curve at 25.0 m/s.
(b) Assuming an unbanked curve, find the minimum force of friction , also called traction, due to the tires and the road.
Strategy and Solution for (a)
We know that [latex]\boldsymbol{{F}_{\textbf{c}}=\frac{mv^2}{r}}.[/latex]Thus,
Strategy for (b)
Figure 2 shows the forces acting on the car on an unbanked (level ground) curve. Friction is to the left, keeping the car from slipping, and because it is the only horizontal force acting on the car, the friction is the centripetal force in this case.
We could also solve part (a) using the first expression in [latex]\begin{array}{l} \boldsymbol{F_{\textbf{c}}=m\frac{v^2}{r}} \\ \boldsymbol{F_{\textbf{c}}=mr\omega^2} \end{array}[/latex][latex]\rbrace[/latex], because m, v, and r are given.
TAKE-HOME EXPERIMENT
Ask a friend or relative to swing a golf club or a tennis racquet. Take appropriate measurements to estimate the centripetal acceleration of the end of the club or racquet. You may choose to do this in slow motion.
PHET EXPLORATIONS: GRAVITY AND ORBITS
Move the sun, earth, moon and space station to see how it affects their gravitational forces and orbital paths. Visualize the sizes and distances between different heavenly bodies, and turn off gravity to see what would happen without it!
Section Summary
- Centripetal force Fc is any force causing uniform circular motion. It is a “centre-seeking” force that always points toward the center of rotation. It is perpendicular to linear velocity v and has mag