Centripetal Acceleration 13 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 18 0. What should the maximum speed be so
that any vehicle can manage the corner even if there is no
friction? 18 0 200 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 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?
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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 Cassiopia takes a ride on childs Ferris Wheel. This
ride has no retaining bar, so that she only sides on the seat as
the ride moves. a)Determine the Normal Force she would experience
from the bottom of the seat when she is at the lowest point on the
ride. b)Determine the Normal Force she would experience from the
bottom of the seat when she is at the highest point on the ride.
c)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.
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Cassiopia takes a ride on childs Ferris Wheel. This ride has no
retaining bar, so that she only sides on the seat as the ride
moves. a)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.
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Cassiopia takes a ride on childs 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.
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Example 12 Cassiopia takes a ride on childs Ferris Wheel. This
ride has no retaining bar, so that she only sides on the seat as
the ride moves. Determine the Net Force (force of seat) 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.
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Example 13 (Hard Question) An engineer has design a banked
corner with a radius of R and an angle of . What is the equation
that determines the velocity of the car given that the coefficient
of friction is ?
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Example 13 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 13 Step 2
(Components) +y +x+x From Vertical
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Sub into Horizontal Example 13 Step 2 (Components) +y +x+x From
Vertical Solve for v Minimum velocity (slides down)