# 6.3 Centripetal force

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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

${a}_{\text{c}}=\frac{{v}^{2}}{r}$

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 $\omega$ , then we can use

${a}_{\text{c}}=r{\omega }^{2}.$

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: ${F}_{\text{net}}=ma.$ For uniform circular motion, the acceleration is the centripetal acceleration: . $a={a}_{\text{c}}.$ Thus, the magnitude of centripetal force ${F}_{\text{c}}$ is

${F}_{\text{c}}=m{a}_{\text{c}}.$

By substituting the expressions for centripetal acceleration ${a}_{\text{c}}$ $\left({a}_{\text{c}}=\frac{{v}^{2}}{r};{a}_{\text{c}}=r{\omega }^{2}\right),$ we get two expressions for the centripetal force ${F}_{\text{c}}$ in terms of mass, velocity, angular velocity, and radius of curvature:

${F}_{\text{c}}=m\frac{{v}^{2}}{r};\phantom{\rule{0.5em}{0ex}}{F}_{\text{c}}=mr{\omega }^{2}.$

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

$r=\frac{m{v}^{2}}{{F}_{\text{c}}}.$

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] .

## 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] ).

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It is the study of the electromagnetic force, one of the four fundamental forces of nature. ... It includes the electric force, which pushes all charged particles, and the magnetic force, which only pushes moving charges.
Energy
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units as in how
praise
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F = m x a
Santos
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can u tell me I cant remember
Indigo
force is equal to mass times acceleration
Santos
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David
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Manorama
To determine the exact value of a percent of a given quantity we need to express the given percent as fraction and multiply it by the given number.
AMIT
meaning
Winford
briefly discuss rocket in physics
ok let's discuss
Jay
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physics is the study of natural phenomena with concern with matter and energy and relationships between them
Ibrahim
a potential difference of 10.0v is connected across a 1.0AuF in an LC circuit. calculate the inductance of the inductor that should be connected to the capacitor for the circuit to oscillate at 1125Hza potential difference of 10.0v is connected across a 1.0AuF in an LC circuit. calculate the inducta
L= 0.002H
NNAEMEKA
how did you get it?
Favour
is the magnetic field of earth changing
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tibebeab
can you explain it
Guhan
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tibebeab
and not every thing have to pop into existence. and it could have always been there . and some scientists think that energy might have been the only entity in the euclidean(imaginary time T=it) which is time undergone wick rotation.
tibebeab
What is projectile?
An object that is launched from a device
Grant
2 dimensional motion under constant acceleration due to gravity
Awais
Not always 2D Awais
Grant
Awais
why not? a bullet is a projectile, so is a rock I throw
Grant
bullet travel in x and y comment same as rock which is 2 dimensional
Awais
components
Awais
no all pf you are wrong. projectile is any object propelled through space by excretion of a force which cease after launch
tibebeab
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tibebeab
it is the object not the motion or its components
tibebeab
where are body center of mass on present.
on the mid point
Suzana
is the magnetic field of the earth changing?
tibebeab
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tibebeab
and for the question from bal want do you mean human body or just any object in space
tibebeab
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Kyla
In this case, the velocity of sound is 350 m/s.
Zahangir
why?
Kyla
some calculations is need. then you will get exact result.
Zahangir
i mean how? isn't it just a d over t?
Kyla
calculate the time it takes the stone to hit the ground then minus the stone's time to the total time... then divide the total distance by the difference of the time
Snuggly
awit lenard. Hahahah ari ga to!
Kyla