11.3. Practice Sets: Circular Motion
Chapter Equations
[latex]\theta=\frac{s}{r}\text{ [rad]}[/latex] (1)
[latex]\omega=\frac{\theta}{t}\text{ [rad/s]}[/latex] (2)
[latex]\omega=\frac{v}{r}[/latex] or [latex]v=\omega r\text{ }[\frac{m}{s}][/latex] (3)
[latex]f=\frac{1}{T}\text{ [Hz]}[/latex] (4)
[latex]\omega=2\pi f[/latex] or [latex]\omega=\frac{2\pi}{T}\text{ [rad/s]}[/latex] (5)
[latex]a_c=\frac{v^2}{r} [m/s^2][/latex] (6)
[latex]F_c=m\times\frac{v^2}{r}\text{ [N]}[/latex] (7)
[latex]\alpha=\frac{\omega_f-\omega_i}{t} [rad/s^2][/latex] (8)
[latex]\theta=\omega_it+\frac{1}{2}\alpha t^2\text{ [rad]}[/latex] (9)
[latex]\theta=\frac{\omega_i+\omega_f}{2}\times t\text{ [rad]}[/latex] (10)
[latex]\omega_f^2=\omega_i^2+2\alpha\theta[/latex] (11)
Try It!
- A cutting disc slows down to a rest from a high angular velocity of [latex]540 \text { rad/s}[/latex]. True or false?
- This is an accelerated circular motion. [latex](T/F)[/latex]
- The angular acceleration is positive. [latex](T/F)[/latex]
- If it takes a flywheel 10 seconds to complete one rotation, then the information should be interpreted as (circle the correct answer):
- The frequency of the motion is [latex]10 s[/latex].
- The period of the motion is [latex]10 s[/latex].
- The total time the flywheel was in motion is [latex]10 s[/latex].
- If a gear completes 8 full rotations at a constant angular velocity, then the information should be interpreted as (circle the correct answer):
- The displacement of the gear is [latex]8 \text{ rad}[/latex].
- The displacement of the gear is approximately [latex]50.3 \text{ rad}[/latex].
- The angular velocity of the gear is [latex]8 \text{ rad/s}[/latex].
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- The centripetal acceleration is defined only for an accelerated circular motion. [latex](T/F)[/latex]
- The direction of the centripetal acceleration is tangential to the circular path of the object. [latex](T/F)[/latex]
- The centripetal acceleration has the same units as the angular acceleration. [latex](T/F)[/latex]
- The centripetal acceleration is directly proportional to the square of the tangential speed and inversely proportional to the radius of the circle. [latex](T/F)[/latex]
Practice
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A rotating flywheel of diameter [latex]40.0 \text{ cm}[/latex] uniformly accelerates from rest to [latex]250 \text{ rad/s}[/latex] in [latex]15.0 \text{ s}[/latex].
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Find its angular acceleration.
Answer:[latex]16.67\text{ rad/s}^2[/latex] -
Find the linear velocity of a point on the rim of the wheel after [latex]15.0 \text{ s}[/latex].
Answer:[latex]50\text{ m/s}[/latex] -
How many revolutions did the wheel make during the [latex]15.0 \text{ s}[/latex]?
Answer:[latex]298.4\text{ rev}[/latex]
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Find its angular acceleration.
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A [latex]1.00\text{ kg}[/latex] rock is swung at the end of a rope in a horizontal circle of radius [latex]1.50 \text{ m}[/latex]. If one complete revolution takes [latex]0.80 \text{ s}[/latex]:
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What is the speed of the rock?
Answer:[latex]11.8 \text{ m/s}[/latex] -
What is the centripetal acceleration of the rock?
Answer:[latex]92.4 \text{ m/s}^2[/latex] -
What is the centripetal force exerted on the rock?
Answer:[latex]92.4 \text{ N}[/latex]
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What is the speed of the rock?
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A wheel rotates through an angle of [latex]13.8 \text { rad}[/latex] as it slows down from [latex]22.0 \text{ rad/s}[/latex] to [latex]13.5 \text { rad/s}[/latex]. What is the magnitude of the angular acceleration of the wheel?
Answer:[latex]-10.9\text {rad/s}^2[/latex] -
A wheel that is rotating at [latex]33.3 \text{ rad/s}[/latex] is given an angular acceleration of [latex]2.15 \text{ rad/s}[/latex]. Through what angle has the wheel turned when its angular speed reaches [latex]72.0 \text{ rad/s}[/latex]?
Answer:[latex]947.7 \text {rad}[/latex] -
A flywheel turns at [latex]320 \text{ rpm}[/latex].
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What is the angular speed at any point on the wheel?
Answer:[latex]33.5 \text {rad/s}[/latex] -
What is the tangential speed [latex]25.0 \text{ cm}[/latex] from the centre?
Answer:[latex]8.4 \text{ m/s}[/latex]
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What is the angular speed at any point on the wheel?
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Find the time needed for the wheel to go from [latex]10.0 \text{ rev/s}[/latex] to [latex]22.0 \text{ rev/s}[/latex].
Answer:[latex]2.5 \text{ s}[/latex] -
A race car, moving at a tangential speed of [latex]60 \text{ m/s}[/latex], covers one full track length in [latex]50 \text{ s}[/latex]. What is the centripetal acceleration of the car?
Answer:[latex]7.5 \text{ m/s}^2[/latex] -
An object moving on a circular path with a radius of [latex]0.02 \text{ m}[/latex], has a centripetal acceleration of [latex]13 \text{ m/s}^2[/latex]. Find the frequency of the motion.
Answer:[latex]4.1 \text{ hz}[/latex]
Challenge Question(s)
Text Version of the Challenge Question(s)
- If a Ferris wheel has height of [latex]100 \text{ m}[/latex], find the angular velocity in rotations per minute if the riders in the carts are going [latex]7 \text{ m/s}[/latex].
Answer: [latex]1.34\text{ rpm}[/latex]
- Large freight trains accelerate very slowly. Suppose one such train accelerates from rest, giving its [latex]0.35\text{ m}[/latex] radius wheels an angular acceleration of [latex]0.25\text{ rad/s}^2[/latex]. After the wheels have made [latex]200[/latex] revolutions (assume no slippage):
- How far has the train moved down the track?
Answer: [latex]439.8\text{ m}[/latex] - What are the final angular velocity of the wheels and the linear velocity of the train?
Answer: [latex]25.07\text{ rad/s}, 8.77\text{ m/s}[/latex]
- How far has the train moved down the track?
- A toy plane is suspended from a string and flying in a horizontal circle. The mass of the plane is [latex]631\text{ g}[/latex] and the plane makes a complete circle every [latex]2.15\text{ seconds}[/latex]. The radius of the circle is [latex]0.950\text{ m}[/latex]. Find:
- The velocity of the plane.
Answer: [latex]2.78\text{ m/s}[/latex] - The acceleration of the plane.
Answer: [latex]8.11\text{ m/s}^2[/latex] - The net force actin on the plane.
Answer: [latex]5.12\text{ N}[/latex]
- The velocity of the plane.
- During a gust of wind, the blades of the windmill are given an angular acceleration of [latex]0.4\text{ rad/s}^2[/latex]. If initially the blades have an angular velocity of [latex]4.0\text{ rad/s}[/latex], determine the speed (in [latex]\text{ft/s}[/latex] of point located at the tip of one of the blades just after the blade has turned three revolutions. The length of the blades is [latex]3.0\text{ ft}[/latex].
Answer: [latex]16.7\text{ ft/s}[/latex]
