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Centripetal Acceleration Formula - Definition, Solved Examples

Centripetal Acceleration Formula is Centripetal acceleration is directed towards the center of the circle, while centrifugal acceleration is the apparent outward force experienced by an object in circular motion.
authorImageMurtaza Mushtaq25 Oct, 2023
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Centripetal Acceleration Formula

Definition of Centripetal Acceleration Formula

Centripetal Acceleration Formula: Centripetal acceleration is a fundamental concept in physics that explains the acceleration of an object moving in a circular path. Centripetal acceleration is the acceleration directed towards the center of a circular path. It is essential for any object undergoing circular motion, ensuring it stays on the curved trajectory.

Units of Centripetal Acceleration:

Centripetal acceleration is measured in meters per second squared (m/s²) in the International System of Units (SI).

Centripetal Acceleration Formula

The formula for centripetal acceleration is given by:

a = (v 2 ) / r

Where: - a is the centripetal acceleration. - v is the velocity of the object in the circular path. - r is the radius of the circular path. This formula quantifies how fast an object is changing its direction while moving in a circular path. The acceleration is directly proportional to the square of the velocity and inversely proportional to the radius.

Solved Questions Of Centripetal Acceleration Formula

Let's work through a few problems to better understand centripetal acceleration. Problem 1: An object is moving in a circle with a radius of 5 meters at a speed of 10 m/s. Calculate its centripetal acceleration. Solution: Using the formula: a = (v^2) / r a = (10^2) / 5 a = 100 / 5 a = 20 m/s² Problem 2: A car is moving in a circular track with a radius of 50 meters. If its centripetal acceleration is 8 m/s², what is its speed? Solution: Rearrange the formula to solve for velocity (v): v = √(a * r) v = √(8 * 50) v = √400 v = 20 m/s Problem 3: A satellite is in orbit around the Earth with a radius of 6,500 kilometers. If its centripetal acceleration is 9.8 m/s², what is its orbital velocity? Solution: v = √(a * r) v = √(9.8 * 6,500,000) v ≈ 2,541.48 m/s Problem 4: A bicycle tire has a radius of 0.4 meters. If it's moving at a speed of 5 m/s, calculate its centripetal acceleration. Solution: a = (v^2) / r a = (5^2) / 0.4 a ≈ 62.5 m/s² Problem 5: A car travels around a curved road with a radius of 100 meters at a constant speed of 20 m/s. What is the car's centripetal acceleration? Solution: a = (v^2) / r a = (20^2) / 100 a = 400 / 100 a = 4 m/s² Problem 6: A Ferris wheel has a radius of 15 meters and takes 2 minutes to complete one rotation. Calculate the centripetal acceleration of a passenger at the highest point of the ride. Solution: First, find the angular velocity (ω) using the time taken for one rotation: ω = (2π) / (2 * 60) ≈ 0.0524 rad/s Now, use the formula for centripetal acceleration: a = ω^2 * r a ≈ (0.0524^2) * 15 a ≈ 0.04 m/s² Problem 7: A stone is tied to a string and is swung around in a circle with a radius of 2 meters. If the stone has a speed of 10 m/s, what is its centripetal acceleration? Solution: a = (v^2) / r a = (10^2) / 2 a = 100 / 2 a = 50 m/s² Problem 8: A small airplane is flying in a circular path with a radius of 500 meters at a speed of 120 m/s. Determine its centripetal acceleration. Solution: a = (v^2) / r a = (120^2) / 500 a = 14,400 / 500 a = 28.8 m/s² Problem 9: A child is spinning a yo-yo in a circle with a radius of 0.3 meters. If the yo-yo's speed is 2 m/s, what is its centripetal acceleration? Solution: a = (v^2) / r a = (2^2) / 0.3 a ≈ 13.33 m/s² Problem 10: A car is driving around a circular racetrack with a radius of 200 meters. If its centripetal acceleration is 6 m/s², calculate its speed. Solution: Rearrange the formula to solve for velocity (v): v = √(a * r) v = √(6 * 200) v ≈ √1200 v ≈ 34.64 m/s Problem 11: A planet orbits around a star with a radius of 1.5 x 10^11 meters. If its orbital velocity is 30,000 m/s, find the planet's centripetal acceleration. Solution: a = (v^2) / r a = (30,000^2) / (1.5 x 10^11) a = 900,000,000 / (1.5 x 10^11) a = 6,000,000 m/s² Problem 12: A rotating space station has a radius of 200 meters. If it rotates at a rate of 0.1 revolutions per minute, calculate the centripetal acceleration at its outer rim. Solution: First, find the angular velocity (ω) using the rate of rotation: ω = (2π * 0.1) / 60 ≈ 0.01047 rad/s Now, use the formula for centripetal acceleration: a = ω^2 * r a ≈ (0.01047^2) * 200 a ≈ 0.0022 m/s²

Centripetal Acceleration Formula FAQs

What is the key difference between centripetal and centrifugal acceleration?

Centripetal acceleration is directed towards the center of the circle, while centrifugal acceleration is the apparent outward force experienced by an object in circular motion. It's important to note that centrifugal acceleration is not a real force but rather a perceived effect due to inertia.

Can centripetal acceleration be negative?

No, centripetal acceleration is always positive. It represents the acceleration directed towards the center of the circular path. In physics, direction is considered when determining the sign of acceleration, and centripetal acceleration is always in a positive direction, pointing inward.

How does centripetal acceleration relate to gravity?

In situations like the Earth's orbit, centripetal acceleration due to gravity keeps objects in orbit around the planet. The gravitational force provides the necessary centripetal acceleration to counteract the tendency of the object to move in a straight line.

Is centripetal acceleration responsible for the feeling of being pushed against the car door during a sharp turn?

Yes, during a turn, the car's centripetal acceleration pushes passengers towards the outer side of the turn. This sensation is a result of centripetal acceleration working to keep the car moving in a curved path while passengers tend to move in a straight line due to inertia.
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