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By the end of the section, you will be able to:
  • Explain the equation for centripetal acceleration
  • Apply Newton’s second law to develop the equation for centripetal force
  • Use circular motion concepts in solving problems involving Newton’s laws of motion

In Motion in Two and Three Dimensions , we examined the basic concepts of circular motion. An object undergoing circular motion, like one of the race cars shown at the beginning of this chapter, must be accelerating because it is changing the direction of its velocity. We proved that this centrally directed acceleration, called centripetal acceleration    , is given by the formula

a c = v 2 r

where v is the velocity of the object, directed along a tangent line to the curve at any instant. If we know the angular velocity ω , then we can use

a c = r ω 2 .

Angular velocity gives the rate at which the object is turning through the curve, in units of rad/s. This acceleration acts along the radius of the curved path and is thus also referred to as a radial acceleration.

An acceleration must be produced by a force. 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: F net = m a . For uniform circular motion, the acceleration is the centripetal acceleration: . a = a c . Thus, the magnitude of centripetal force F c is

F c = m a c .

By substituting the expressions for centripetal acceleration a c ( a c = v 2 r ; a c = r ω 2 ) , we get two expressions for the centripetal force F c in terms of mass, velocity, angular velocity, and radius of curvature:

F c = m v 2 r ; F c = m r ω 2 .

You may use whichever expression for centripetal force is more convenient. Centripetal force F c is always perpendicular to the path and points to the center of curvature, because a c is perpendicular to the velocity and points to the center of curvature. Note that if you solve the first expression for r , you get

r = m v 2 F c .

This implies that for a given mass and velocity, a large centripetal force causes a small radius of curvature—that is, a tight curve, as in [link] .

The figure consists of two semicircles. The semicircle on the left has radius r and bigger than the one on the right, which has radius r prime. In both the figures, the direction of the motion is given as counter-clockwise along the semicircles. A point is shown on the path, where the radius is shown with an arrow pointing out from the center of the semicircle. At the same point, the centripetal force, F sub c, is shown pointing inward, in the opposite direction to that of radius arrow. The velocity, v, is shown at this point as well, and it is tangent to the semicircle, pointing left and up, perpendicular to the forces. In both the figures, the velocity is same, but the radius prime is smaller and centripetal force is larger in the figure on the right. It is noted that vector F sub c is parallel to vector a sub c since vector F sub c equals m times vector a sub c.
The frictional force supplies the centripetal force and is numerically equal to it. Centripetal force is perpendicular to velocity and causes uniform circular motion. The larger the F c , the smaller the radius of curvature r and the sharper the curve. The second curve has the same v , but a larger F c produces a smaller r ′.

What coefficient of friction do cars need on a flat curve?

(a) Calculate the centripetal force exerted on a 900.0-kg car that negotiates a 500.0-m radius curve at 25.00 m/s. (b) Assuming an unbanked curve, find the minimum static coefficient of friction between the tires and the road, static friction being the reason that keeps the car from slipping ( [link] ).

Questions & Answers

what is the formula for pressure?
Goodness Reply
force/area
Kidus
force is newtom
Kidus
and area is meter squared
Kidus
so in SI units pressure is N/m^2
Kidus
In customary United States units pressure is lb/in^2. pound per square inch
Kidus
who is Newton?
John Reply
scientist
Jeevan
a scientist
Peter
that discovered law of motion
Peter
ok
John
but who is Isaac newton?
John
a postmodernist would say that he did not discover them, he made them up and they're not actually a reality in itself, but a mere construct by which we decided to observe the word around us
elo
how?
Qhoshe
what is a scalar quantity
Peter Reply
scalar: are quantity have numerical value
muslim
is that a better way in defining scalar quantity
Peter
thanks
muslim
quantity that has magnitude but no direction
Peter
upward force and downward force lift
adegboye Reply
upward force and downward force on lift
adegboye
hi
Etini
hi
elo
hy
Xander
Hello
Jux_dob
hi
Peter
Helo
Tobi
what's the answer? I can't get it
Rachel Reply
what is the question again?
Sallieu
What's this conversation?
Zareen
what is catenation? and give examples
sununu
How many kilometres in 1 mile
Nessy
1.609km in 1mile
Faqir
what's the si unit of impulse
Iguh Reply
The Newton second (N•s)
Ethan
what is the s. I unit of current
Roland Reply
Amphere(A)
imam
thanks man
Roland
u r welcome
imam
the velocity of a boat related to water is 3i+4j and that of water related to earth is i-3j. what is the velocity of the boat relative to earth.If unit vector i and j represent 1km/hour east and north respectively
Pallavi Reply
what is head to tail rule?
kinza Reply
Explain Head to tail rule?
kinza
what is the guess theorem
Monu Reply
viva question and answer on practical youngs modulus by streching
Akash Reply
send me vvi que
rupesh
a car can cover a distance of 522km on 36 Liter's of petrol, how far can it travel on 14 liter of petrol.
Isaac
yoo the ans is 193
Joseph
whats a two dimensional force
Jimoh Reply
what are two dimensional force?
Ahmad
Where is Fourier Theorem?
Atul Reply
what is Boyle's law
Amoo Reply
Boyle's law state that the volume of a given mass of gas is inversely proportion to its pressure provided that temperature remains constant
Abe
how do I turn off push notifications on this crap app?
Huntergirl
what is the meaning of in.
CHUKWUMA Reply
In means natural logarithm
Elom
is dea graph for cancer caliper experiment using glass block?
Bako
why the ship created in search way?
Mershak
Practice Key Terms 6

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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