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8.2 Centripetal acceleration  (Page 3/6)

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Of course, a net external force is needed to cause any acceleration, just as Newton proposed in his second law of motion. So a net external force is needed to cause a centripetal acceleration. In Centripetal Force , we will consider the forces involved in circular motion.

Phet explorations: ladybug motion 2d

Learn about position, velocity and acceleration vectors. Move the ladybug by setting the position, velocity or acceleration, and see how the vectors change. Choose linear, circular or elliptical motion, and record and playback the motion to analyze the behavior.

Ladybug Motion 2D

Section summary

  • Centripetal acceleration a c size 12{a rSub { size 8{c} } } {} is the acceleration experienced while in uniform circular motion. It always points toward the center of rotation. It is perpendicular to the linear velocity v size 12{v} {} and has the magnitude
    a c = v 2 r ; a c = rω 2 . size 12{a rSub { size 8{c} } = { {v rSup { size 8{2} } } over {r} } `; a rSub { size 8{c} } =rω rSup { size 8{2} } } {}
  • The unit of centripetal acceleration is m / s 2 size 12{m/s rSup { size 8{2} } } {} .

Conceptual questions

Can centripetal acceleration change the speed of circular motion? Explain.

Problem exercises

A fairground ride spins its occupants inside a flying saucer-shaped container. If the horizontal circular path the riders follow has an 8.00 m radius, at how many revolutions per minute will the riders be subjected to a centripetal acceleration whose magnitude is 1.50 times that due to gravity?

12.9 rev/min

A runner taking part in the 200 m dash must run around the end of a track that has a circular arc with a radius of curvature of 30 m. If he completes the 200 m dash in 23.2 s and runs at constant speed throughout the race, what is the magnitude of his centripetal acceleration as he runs the curved portion of the track?

Taking the age of Earth to be about 4 × 10 9 size 12{4 times "10" rSup { size 8{9} } } {} years and assuming its orbital radius of 1.5 × 10 11 size 12{4 times "10" rSup { size 8{9} } } {} has not changed and is circular, calculate the approximate total distance Earth has traveled since its birth (in a frame of reference stationary with respect to the Sun).

4 × 10 21 m size 12{ {underline {4 times "10" rSup { size 8{"21"} } " m"}} } {}

The propeller of a World War II fighter plane is 2.30 m in diameter.

(a) What is its angular velocity in radians per second if it spins at 1200 rev/min?

(b) What is the linear speed of its tip at this angular velocity if the plane is stationary on the tarmac?

(c) What is the centripetal acceleration of the propeller tip under these conditions? Calculate it in meters per second squared and convert to multiples of g .

An ordinary workshop grindstone has a radius of 7.50 cm and rotates at 6500 rev/min.

(a) Calculate the magnitude of the centripetal acceleration at its edge in meters per second squared and convert it to multiples of g .

(b) What is the linear speed of a point on its edge?

a) 3 . 47 × 10 4 m / s 2 size 12{3 cdot "47" times "10" rSup { size 8{"4"} } m/s rSup { size 8{2} } } {} , 3 . 55 × 10 3 g size 12{3 cdot "55" times "10" rSup { size 8{"3"} } g} {}

b) 51 . 1 m / s size 12{3 cdot "75" m/s} {}

Helicopter blades withstand tremendous stresses. In addition to supporting the weight of a helicopter, they are spun at rapid rates and experience large centripetal accelerations, especially at the tip.

(a) Calculate the magnitude of the centripetal acceleration at the tip of a 4.00 m long helicopter blade that rotates at 300 rev/min.

(b) Compare the linear speed of the tip with the speed of sound (taken to be 340 m/s).

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Source:  OpenStax, College physics arranged for cpslo phys141. OpenStax CNX. Dec 23, 2014 Download for free at http://legacy.cnx.org/content/col11718/1.4
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