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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 12.2 Centripetal Acceleration

Subsubsection Warm-up Activities

Activity 12.2.1. Exploration - Curved Road.

Suppose you are in the right seat of a car driving along a curved road. Use everyday language to explain what happens as the car is turning to the left along a curved road. In particular:
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  1. What direction do you move relative to the car?
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  2. What direction do you move relative to the ground?
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  3. What do you feel as the car turns?
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Activity 12.2.2. Prediction - Car Turning.

A car moves counterclockwise at constant speed along the curved road shown below.
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A top-view diagram of a road curving from the bottom of the screen to the top left.
Figure 12.2.1. A car moving along a curved road.
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Choose an origin and coordinates and use them to sketch vectors you believe best represent the directions of the (a) instantaneous position, (b) instantaneous velocity, and (c) instantaneous acceleration of the car at the instant shown. Explain how you predicted the direction of each vector.
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Activity 12.2.3. Prediction - Car Slowing Down.

Suppose the car in the previous activity were instead slowing down. How if at all would you expect the three vectors to change?
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Compare your predictions to the solutions in the video below.
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Subsubsection Key Ideas

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When an object moves along a curved path, the instantaneous velocity always points tangent to the curve.
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Subsubsection Model Applications

Activity 12.2.4. Acceleration Direction.

The figure below shows four different instants (A-D) for a car moving clockwise along a curved path at constant speed.
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Three cases, each showing a car moving clockwise around a curved path, with points A and B far apart, A and C closer together, and A and D very close together.
Figure 12.2.3.
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(a)
Sketch vectors for the instantaneous velocity at each instant.
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(b)
Sketch vectors for the change in velocity for each case.
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(c)
Use the definition of instantaneous acceleration to determine the direction of the car’s acceleration at instant A.
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Activity 12.2.5. Acceleration Magnitude.

The figure below shows three identical cars, each moving around a circular track. Cars A and B move at different constant speeds around tracks with the same radius. Cars A and C move at the same speed around tracks with different radii.
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Three cases, each showing a car moving clockwise around a circular path of radius d. Car A moves at constant speed v, Car B moves at constant speed 2v, and Car C moves at constant speed v around a circle of radius d/2.
Figure 12.2.5.
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(a)
Predict which of the three cars has an acceleration with the largest magnitude. Explain your reasoning
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(b)
Sketch vectors for the instantaneous velocity for each point.
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(c)
Sketch vectors for the change in velocity for each case.
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(d)
Use the definition of instantaneous acceleration to rank the magnitudes of the cars’ accelerations.
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Definition 12.2.8. Centripetal Acceleration.

An object moving along a curved path experiences a centripetal acceleration that points perpendicular to the path (inward) with magnitude \(a_{c} = \frac{v^2}{r}\text{,}\) where \(r\) is the local radius of curvature. If the object is moving at constant speed, the acceleration is entirely centripetal.
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