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Centripetal Acceleration 12 Examples with full solutions

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Example 1 A 1500 kg car is moving on a flat road and negotiates a curve whose radius is 35m. If the coefficient of static friction between the tires and the road is 0.5, determine the maximum speed the car can have in order to successfully make the turn. 35m

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Example 1 – Step 1 (Free Body Diagram) +y +x This static friction is the only horizontal force keeping the car moving toward the centre of the arc (else the car will drive off the road). Acceleration direction

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Example 1 - Step 2 (Sum of Vector Components ) +y +x Vertical ComponentsHorizontal Components We have an acceleration in x-direction Static Friction From Vertical Component

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Example 1 - Step 3 (Insert values) +y +x

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Example 2 A car is travelling at 25m/s around a level curve of radius 120m. What is the minimum value of the coefficient of static friction between the tires and the road to prevent the car from skidding? 120 m

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Example 2 – Step 1 (Free Body Diagram) +y +x This static friction is the only horizontal force keeping the car moving toward the centre of the arc (else the car will drive off the road). Acceleration direction

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Example 2 - Step 2 (Sum of Vector Components ) +y +x Vertical ComponentsHorizontal Components We have an acceleration in x-direction Static Friction From Vertical Component

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Example 2 - Step 3 (Insert values) +y +x We require the minimum value

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Example 3 An engineer has design a banked corner with a radius of 200m and an angle of What should the maximum speed be so that any vehicle can manage the corner even if there is no friction? m

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Example 3 – Step 1 (Free Body Diagram) First the car Now for gravity The normal to the road Components of Normal force along axis (we ensured one axis was along acceleration direction +y +x Acceleration direction Notice that we have no static friction force in this example (question did not require one)

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Example 3 - Step 2 (Sum of Vector Components ) +y +x Vertical Components Horizontal Components We have an acceleration in the x-direction From Vertical Component

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Example 3 - Step 3 (Insert values) +y +x

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Example 4 An engineer has design a banked corner with a radius of 230m and the bank must handle speeds of 88 km/h. What bank angle should the engineer design to handle the road if it completely ices up? 230 m ?

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Example 4 – Step 1 (Free Body Diagram) First the car Now for gravity The normal to the road Components of Normal force along axis (we ensured one axis was along acceleration direction +y +x Acceleration direction Notice that we have no static friction force in this example (question did not require one)

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Example 4 - Step 2 (Sum of Vector Components ) +y +x Vertical Components Horizontal Components We have an acceleration in the x-direction From Vertical Component

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+y +x Example 4 - Step 3 (Insert values) Dont forget to place in metres per second

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Example 5 A 2kg ball is rotated in a vertical direction. The ball is attached to a light string of length 3m and the ball is kept moving at a constant speed of 12 m/s. Determine the tension is the string at the highest and lowest points.

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Example 5 – Step 1 (Free Body Diagram) Top Bottom When the ball is at the top of the curve, the string is pulling down. When the ball is at the bottom of the curve, the string is pulling up. In both cases, gravity is pulling down Note: any vertical motion problems that do not include a solid attachment to the centre, do not maintain a constant speed, v and thus (except at top and bottom) have an acceleration that does not point toward the centre. (it is better to use energy conservation techniques)

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Example 5 - Step 2 (Sum of Vector Components ) Top Bottom +y +x Note: acceleration is down (-)

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Example 5 - Step 3 (Insert values) +y +x Top Bottom

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Example 6 A conical pendulum consists of a mass (the pendulum bob) that travels in a circle on the end of a string tracing out a cone. If the mass of the bob is 1.7 kg, and the length of the string is 1.25 m, and the angle the string makes with the vertical is 25 o. Determine: a) the speed of the bob b) the frequency of the bob

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Example 6 – Step 1 (Free Body Diagram) Lets decompose our Tension Force into vertical and horizontal components +y +x Its easier to make the x axis positive to the left

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+y +x Example 6 - Step 2 (Sum of Vector Components ) Horizontal Vertical

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Example 6 - Step 3 (Insert values for velocity) +y +x The speed of the bob is about 1.55 m/s

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Example 6 - Step 3 (Insert values for frequency) +y +x The frequency of the bob is about 0.468Hz

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Example 7 A swing at an amusement park consists of a vertical central shaft with a number of horizontal arms. Each arm supports a seat suspended from a cable 5.00m long. The upper end of the cable is attached to the arm 3.00 m from the central shaft. Determine the time for one revolution of the swing if the cable makes an angle of 30 0 with the vertical

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Example 7 – Step 1 (Free Body Diagram) +y +x

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Example 7 - Step 2 (Sum of Vector Components ) +y +x Horizontal Vertical

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Example 7 - Step 2 (Sum of Vector Components ) +y +x The period is 6.19s

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Example 8 A toy car with a mass of 1.60 kg moves at a constant speed of 12.0 m/s in a vertical circle inside a metal cylinder that has a radius of 5.00 m. What is the magnitude of the normal force exerted by the walls of the cylinder at A the bottom of the circle and at B the top of the circle

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Example 8 – Step 1 (Free Body Diagram) Top Bottom When the car is at the top of the curve, the normal force is pushing down. When the ball is at the bottom of the curve, the normal force is pushing up. In both cases, gravity is pulling down Note: any vertical motion problems that do not include a solid attachment to the centre, do not maintain a constant speed, v and thus (except at top and bottom) have an acceleration that does not point toward the centre. (it is better to use energy conservation techniques)

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Example 8 - Step 2 (Sum of Vector Components ) Top Bottom +y +x Note: acceleration is down (-)

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Example 8 - Step 3 (Insert values) +y +x Top Bottom

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Example 9 A 0.20g fly sits 12cm from the centre of a phonograph record revolving at rpm. a) What is the magnitude of the centripetal force on the fly? b) What is the minimum static friction between the fly and the record to prevent the fly from sliding off?

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Example 8 – Step 1 (Free Body Diagram)

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Example 9 - Step 2 (Sum of Vector Components ) a. Convert to correct units

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Example 9 - Step 2 (Sum of Vector Components ) b.

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Example 10 A 4.00 kg mass is attached to a vertical rod by the means of two 1.25 m strings which are 2.00 m apart. The mass rotates about the vertical shaft producing a tension of 80.0 N in the top string. a)What is the tension on the lower string? b)How many revolutions per minute does the system make?

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Example 10 – Step 1 (Free Body Diagram) +y +x

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Example 10 - Step 2 (Sum of Vector Components) Horizontal Vertical +y +

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Example 10 - Step 2 (Sum of Vector Components ) +y + a)

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Example 10 - Step 2 (Sum of Vector Components ) +y + b)

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Example 11 The moon orbits the Earth in an approximately circular path of radius 3.8 x 10 8 m. It takes about 27 days to complete one orbit. What is the mass of the Earth as obtained by this data?

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Example 11 – Step 1 (Free Body Diagram)

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Example 11 - Step 2 (Sum of Vector Components) The mass of the Earth is about 6.0x10 24 kg Horizontal

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Example 12 (Hard Question) An engineer has design a banked corner with a radius of R and an angle of β. What is the equation that determines the velocity of the car given that the coefficient of friction is µ ?

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Example 12 – Step 1 (Free Body Diagram) First the car Now for gravity The normal to the road Components of Normal force along axis (we ensured one axis was along acceleration direction +y +x Acceleration direction Friction We have friction going down by assuming car wants to slide up. This will provide an equations for the maximum velocity

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Vertical Components Horizontal Components Example 12 – Step 2 (Components) +y +x+x From Vertical

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Sub into Horizontal Example 12 – Step 2 (Components) +y +x+x From Vertical Solve for v π Minimum velocity (slides down)

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