Centripetal Acceleration 13 Examples with full solutions

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

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

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

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

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

Example 2 – Step 1 (Free Body Diagram) 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 +y +x

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

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

Example 3 An engineer has design a banked corner with a radius of 200m and an angle of 180. What should the maximum speed be so that any vehicle can manage the corner even if there is no friction? 180 200 m

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

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

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

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

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

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

Don’t forget to place in metres per second Example 4 - Step 3 (Insert values) +y +x Don’t forget to place in metres per second

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.

Example 5 – Step 1 (Free Body Diagram) 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) Bottom Top When the ball is at the bottom of the curve, the string is “pulling” up. When the ball is at the top of the curve, the string is “pulling” down. In both cases, gravity is pulling down

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

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

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 25o. Determine: a) the speed of the bob b) the frequency of the bob

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

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

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

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

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 300 with the vertical

Example 7 – Step 1 (Free Body Diagram)

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

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

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

Example 8 – Step 1 (Free Body Diagram) 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) Bottom Top When the ball is at the bottom of the curve, the normal force is “pushing” up. When the car is at the top of the curve, the normal force is “pushing” down. In both cases, gravity is pulling down

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

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

Example 9 A 0.20g fly sits 12cm from the centre of a phonograph record revolving at 33.33 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?

Example 8 – Step 1 (Free Body Diagram)

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

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

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. What is the tension on the lower string? How many revolutions per minute does the system make?

Example 10 – Step 1 (Free Body Diagram)

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

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

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

Example 11 The moon orbits the Earth in an approximately circular path of radius 3.8 x 108 m. It takes about 27 days to complete one orbit. What is the mass of the Earth as obtained by this data?

Example 11 – Step 1 (Free Body Diagram)

Example 11 - Step 2 (Sum of Vector Components) Horizontal The mass of the Earth is about 6.0x1024 kg

Example 12 Cassiopia takes a ride on child’s Ferris Wheel. This ride has no retaining bar, so that she only sides on the seat as the ride moves. Determine the Normal Force she would experience from the bottom of the seat when she is at the lowest point on the ride. Determine the Normal Force she would experience from the bottom of the seat when she is at the highest point on the ride. Determine the Net Force she would experience from the bottom of the seat when she is at the mid-point on the ride with herheight equal to the axis.

Cassiopia takes a ride on child’s Ferris Wheel Cassiopia takes a ride on child’s Ferris Wheel. This ride has no retaining bar, so that she only sides on the seat as the ride moves. Determine the Normal Force she would experience from the bottom of the seat when she is at the lowest point on the ride. With this ride, gravity always points down, the normal (seat) force always points up, and the centripetal acceleration is always toward the centre.

Cassiopia takes a ride on child’s Ferris Wheel Cassiopia takes a ride on child’s Ferris Wheel. This ride has no retaining bar, so that she only sides on the seat as the ride moves. Determine the Normal Force she would experience from the bottom of the seat when she is at the highest point on the ride. With this ride, gravity always points down, the normal (seat) force always points up, and the centripetal acceleration is always toward the centre.

Example 12 Cassiopia takes a ride on child’s Ferris Wheel. This ride has no retaining bar, so that she only sides on the seat as the ride moves. Determine the Net Force she would experience from the bottom of the seat when she is at the mid-point on the ride with her height equal to the axis.

Example 13 An air puck of mass 0.5 kg is tied to a string and allowed to revolve in a circle of radius 1.0 m on a frictionless, horizontal table. The other end of the string passes through a hole in the center of the table and a mass of 2.0 kg is tied to it. The suspended mass remains in equilibrium while the puck on the tabletop revolves? a) What is the tension in the string? b) What is the Central force acting on the puck? c) What is the speed of the puck?

Example 13 An air puck of mass 0.5 kg is tied to a string and allowed to revolve in a circle of radius 1.0 m on a frictionless, horizontal table. The other end of the string passes through a hole in the center of the table and a mass of 2.0 kg is tied to it. The suspended mass remains in equilibrium while the puck on the tabletop revolves? a) What is the tension in the string?

Example 13 An air puck of mass 0.5 kg is tied to a string and allowed to revolve in a circle of radius 1.0 m on a frictionless, horizontal table. The other end of the string passes through a hole in the center of the table and a mass of 2.0 kg is tied to it. The suspended mass remains in equilibrium while the puck on the tabletop revolves? b) What is the Central force acting on the puck?

Example 13 An air puck of mass 0.5 kg is tied to a string and allowed to revolve in a circle of radius 1.0 m on a frictionless, horizontal table. The other end of the string passes through a hole in the center of the table and a mass of 2.0 kg is tied to it. The suspended mass remains in equilibrium while the puck on the tabletop revolves? c) What is the speed of the puck?

Example 14 A proposed space station includes living quarters in a circular ring 62.0 m in diameter. At what angular speed should the ring rotate so the occupants feel that they have the same weight as they do on Earth?  How many revolutions per minute is this? What is the velocity of the astronaut in the living quarters?

Example 14 A proposed space station includes living quarters in a circular ring 62.0 m in diameter. At what angular speed should the ring rotate so the occupants feel that they have the same weight as they do on Earth?  How many revolutions per minute is this? What is the velocity of the astronaut in the living quarters?

Example 15 If you dropped and apple when on the space station, how will it fall? Notice that when the apple is released, there is no longer a net force acting on it.  So in the absence of a net force in continues on in the same direction and speed (constant velocity) that it had the instant it was released.  From our point of view the apple moves in a straight line when released.  From the astronaut's point of view (accelerated reference frame) the apple seems to drop in a straight line toward the floor.  It seems to be pulled to the floor as if by gravity.  The astronaut is fooled by what he sees into believing that there is gravity.

Example 15 (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 ?

Acceleration direction Example 15 – Step 1 (Free Body Diagram) Components of Normal force along axis (we ensured one axis was along acceleration direction The normal to the road First the car +y +x Acceleration direction We have friction going down by assuming car wants to slide up. This will provide an equations for the maximum velocity Friction Now for gravity

Example 15 – Step 2 (Components) +y +x Vertical Components Horizontal Components From Vertical

Example 15 – Step 2 (Components) +y +x From Vertical Sub into Horizontal Solve for v Minimum velocity (slides down) π

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