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N sin θ = mv 2 r . size 12{N"sin"θ= { { ital "mv" rSup { size 8{2} } } over {r} } } {}

Because the car does not leave the surface of the road, the net vertical force must be zero, meaning that the vertical components of the two external forces must be equal in magnitude and opposite in direction. From the figure, we see that the vertical component of the normal force is N cos θ size 12{N"cos"θ} {} , and the only other vertical force is the car's weight. These must be equal in magnitude; thus,

N cos θ = mg . size 12{N"cos"θ= ital "mg"} {}

Now we can combine the last two equations to eliminate N size 12{N} {} and get an expression for θ size 12{θ} {} , as desired. Solving the second equation for N = mg / ( cos θ ) size 12{N= ital "mg"/ \( "cos"θ \) } {} , and substituting this into the first yields

mg sin θ cos θ = mv 2 r
mg tan ( θ ) = mv 2 r tan θ = v 2 rg.

Taking the inverse tangent gives

θ = tan 1 v 2 rg (ideally banked curve, no friction). size 12{θ="tan" rSup { size 8{ - 1} } left ( { {v rSup { size 8{2} } } over { ital "rg"} } right )} {}

This expression can be understood by considering how θ size 12{θ} {} depends on v size 12{v} {} and r size 12{r} {} . A large θ size 12{θ} {} will be obtained for a large v size 12{v} {} and a small r size 12{r} {} . That is, roads must be steeply banked for high speeds and sharp curves. Friction helps, because it allows you to take the curve at greater or lower speed than if the curve is frictionless. Note that θ size 12{θ} {} does not depend on the mass of the vehicle.

In this figure, a car from the backside is shown, turning to the left, on a slope angling downward to the left. A point in the middle of the back of the car is shown which shows one downward vector depicting weight, w, and an upward arrow depicting force N, which is a linear line along the car and is at an angle theta with the straight up arrow. The slope is at an angle theta with the horizontal surface below the slope. The force values, N multipliy sine theta equals to centripetal force, the net force on the car and N cosine theta equal to w are given below the car.
The car on this banked curve is moving away and turning to the left.

What is the ideal speed to take a steeply banked tight curve?

Curves on some test tracks and race courses, such as the Daytona International Speedway in Florida, are very steeply banked. This banking, with the aid of tire friction and very stable car configurations, allows the curves to be taken at very high speed. To illustrate, calculate the speed at which a 100 m radius curve banked at 65.0° should be driven if the road is frictionless.


We first note that all terms in the expression for the ideal angle of a banked curve except for speed are known; thus, we need only rearrange it so that speed appears on the left-hand side and then substitute known quantities.


Starting with

tan θ = v 2 rg size 12{"tan"θ= { {v rSup { size 8{2} } } over { ital "rg"} } } {}

we get

v = ( rg tan θ ) 1 / 2 . size 12{v= \( ital "rg""tan"θ \) rSup { size 8{1/2} } } {}

Noting that tan 65.0º = 2.14, we obtain

v = ( 100 m ) ( 9.80 m /s 2 ) ( 2 . 14 ) 1 / 2 = 45.8 m/s.


This is just about 165 km/h, consistent with a very steeply banked and rather sharp curve. Tire friction enables a vehicle to take the curve at significantly higher speeds.

Calculations similar to those in the preceding examples can be performed for a host of interesting situations in which centripetal force is involved—a number of these are presented in this chapter's Problems and Exercises.

Got questions? Get instant answers now!

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!

Gravity and Orbits

Section summary

  • Centripetal force F c size 12{F rSub { size 8{c} } } {} is any force causing uniform circular motion. It is a “center-seeking” force that always points toward the center of rotation. It is perpendicular to linear velocity v size 12{v} {} and has magnitude
    F c = ma c ,

    which can also be expressed as

    F c = m v 2 r or F c = mr ω 2 ,

Questions & Answers

Propose a force standard different from the example of a stretched spring discussed in the text. Your standard must be capable of producing the same force repeatedly.
Giovani Reply
What is meant by dielectric charge?
It's Reply
what happens to the size of charge if the dielectric is changed?
Brhanu Reply
omega= omega not +alpha t derivation
Provakar Reply
u have to derivate it respected to time ...and as w is the angular velocity uu will relace it with "thita × time""
do to be peaceful with any body
Brhanu Reply
the angle subtended at the center of sphere of radius r in steradian is equal to 4 pi how?
Saeed Reply
if for diatonic gas Cv =5R/2 then gamma is equal to 7/5 how?
define variable velocity
Ali Reply
displacement in easy way.
Mubashir Reply
binding energy per nucleon
Poonam Reply
why God created humanity
Manuel Reply
Because HE needs someone to dominate the earth (Gen. 1:26)
why god made humenity
and he to multiply
stuff happens
God plays dice
Is the object in a conductor or an insulator? Justify your answer. whats the answer to this question? pls need help figure is given above
Jun Reply
ok we can say body is electrically neutral ...conductor this quality is given to most metalls who have free electron in orbital d ...but human doesn't have ...so we re made from insulator or dielectric material ... furthermore, the menirals in our body like k, Fe , cu , zn
when we face electric shock these elements work as a conductor that's why we got this shock
how do i calculate the pressure on the base of a deposit if the deposit is moving with a linear aceleration
ximena Reply
why electromagnetic induction is not used in room heater ?
Gopi Reply
What is position?
Amoah Reply
What is law of gravition
sushil Reply
Practice Key Terms 5

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