Centripetal acceleration and force

Whirling overhead a small mass on a stringImagine whirling a small weight around your head on a string. If the string broke, the weight would fly off and no longer move in a circle. This is a consequence of Newton’s first law: A moving body will continue in a straight line if no net force acts on it. What direction will it travel? An object undergoing circular motion has a velocity that is tangential to the circle. If the string were to break, then the object would continue moving in the direction of its velocity—upward in the illustration. Read the text aloud Show Tangential and radial vectors
Why centripetal acceleration points towards the center of the circleWhat keeps the object in circular motion over your head? The string exerts an inward force that causes an inward acceleration that continually bends the object’s direction of motion toward the center of the circle. In the illustration, the velocity of the object changes direction as it moves from point A to point B. The change in velocity is represented by the vector Δv, which is perpendicular to the direction of velocity. Since acceleration is the rate at which an object’s velocity changes, the centripetal acceleration acts in the direction of the vector Δv, which is toward the center of the circle. Centripetal acceleration always points toward the center of the circle and is perpendicular to the object’s velocity. Read the text aloud
How are acceleration and velocity related quantitatively in circular motion? Look again at the illustration above. Between points A and B the object moves a distance d = vt, while the change in its velocity is Δv. The blue- and green-shaded triangles are similar; therefore, the ratio Δv/v is the same as the ratio vt/r. Acceleration is defined as Δv/t; therefore, the centripetal acceleration is given by equation (7.3). Read the text aloud Show More on angular, linear, and tangential velocity
ac  = centripetal acceleration (m/s2)
vt  = tangential speed (m/s)
r  = radius of circle (m)
Centripetal acceleration
Show Interpreting the equation
Whether a force changes a moving object’s speed or its direction depends on the force’s direction. A force in the direction of motion causes the object to change speed. A force perpendicular to the motion causes the object to change its path from a line to a circle—without changing speed. The force (or combination of forces) that points towards the center of a circle and is perpendicular to an object’s motion is called a centripetal force. Some source of centripetal force is required to keep an object in circular motion, whether it is a planet in orbit or a race car going around a curved track. We get the equation for centripetal force by combining equation (7.3) and Newton’s second law. Read the text aloud Show Is the “centrifugal” force real?
(7.4) F c = m v t 2 r
Fc  = centripetal force (N)
m  = mass (kg)
v  = tangential speed (m/s)
r  = radius of circle (m)
Centripetal force
A Ferrari is speeding around a circular track with a radius of 400 m. At a point around the loop, it is clocked at a tangential velocity of 70 m/s. What is the Ferrari’s centripetal acceleration at this moment?
  1. 12.25 m/s2
  2. 0.175 m/s2
  3. 24.50 m/s2
  4. 4.375 m/s2

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