12. Rotational Kinematics

Converting Between Linear & Rotational

12. Rotational Kinematics

Converting Between Linear & Rotational

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Converting Between Linear & Rotational

Length of string rotating point mass

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Relating angular and regular motion variables | Physics | Khan Academy

Rotational Motion: An Explanation, Angular Displacement, Velocity and Acceleration

Angular & Translational Quantities

Jennifer Cash

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Multiple Choice

A disc of radius 10 m rotates around itself with a constant 180 RPM. Calculate the linear speed at a point 7 m from the center of the disc.

A rock rotates around a light, 4-m long string. The rock is initially at rest, but reaches 150 RPM in 3 seconds. Calculate its tangential acceleration after 3 s.

BONUS:Calculate its tangential speed after 3 s.

A 4 m long blade initially at rest begins to spin with 3 rad/s² around its axis, which is located at the middle of the blade. It accelerates for 10 s. Find the tangential speed of a point at the tip of the blade 10 s after it starts rotating.

A bicycle wheel has two clumps of mud stuck to it: Clump A is stuck to the outer edge of the tire, and clump B is stuck to the spokes, 2/3 of the way out from the center. What are the ratios of the speeds and angular speeds of the two clumps:

v_{A} / v_{B}

and

ω_{A} / ω_{B}

428

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Textbook Question

Flywheels—rapidly rotating disks—are widely used in industry for storing energy. They are spun up slowly when extra energy is available, then decelerate quickly when needed to supply a boost of energy. A 20-cm-diameter rotor made of advanced materials can spin at 100,000 rpm. a. What is the speed of a point on the rim of this rotor?

253

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Textbook Question

A Ferris wheel of radius R speeds up with angular acceleration starting from rest. Find expressions for the (a) velocity and (b) centripetal acceleration of a rider after the Ferris wheel has rotated through angle ∆θ.

224

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Textbook Question

A typical laboratory centrifuge rotates at 4000 rpm. Test tubes have to be placed into a centrifuge very carefully because of the very large accelerations. a. What is the acceleration at the end of a test tube that is 10 cm from the axis of rotation?

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Textbook Question

Compact Disc. A compact disc (CD) stores music in a coded pattern of tiny pits 10^-7 m deep. The pits are arranged in a track that spirals outward toward the rim of the disc; the inner and outer radii of this spiral are 25.0 mm and 58.0 mm, respectively. As the disc spins inside a CD player, the track is scanned at a constant linear speed of 1.25 m/s. (c) What is the average angular acceleration of a maximumduration CD during its 74.0-min playing time? Take the direction of rotation of the disc to be positive.

291

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Textbook Question

Compact Disc. A compact disc (CD) stores music in a coded pattern of tiny pits 10^-7 m deep. The pits are arranged in a track that spirals outward toward the rim of the disc; the inner and outer radii of this spiral are 25.0 mm and 58.0 mm, respectively. As the disc spins inside a CD player, the track is scanned at a constant linear speed of 1.25 m/s. (b) The maximum playing time of a CD is 74.0 min. What would be the length of the track on such a maximum-duration CD if it were stretched out in a straight line?

916

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Textbook Question

An electric turntable 0.750 m in diameter is rotating about a fixed axis with an initial angular velocity of 0.250 rev/s and a constant angular acceleration of 0.900 rev/s^2. (c) What is the tangential speed of a point on the rim of the turntable at t = 0.200 s?

A wheel of diameter 40.0 cm starts from rest and rotates with a constant angular acceleration of 3.00 rad/s^2. Compute the radial acceleration of a point on the rim for the instant the wheel completes its second revolution from the relationship (b) a_rad = v^2/r

466

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Textbook Question

Compact Disc. A compact disc (CD) stores music in a coded pattern of tiny pits 10^-7 m deep. The pits are arranged in a track that spirals outward toward the rim of the disc; the inner and outer radii of this spiral are 25.0 mm and 58.0 mm, respectively. As the disc spins inside a CD player, the track is scanned at a constant linear speed of 1.25 m/s. (a) What is the angular speed of the CD when the innermost part of the track is scanned? The outermost part of the track?

200

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Textbook Question

(II) How fast (in rpm) must a centrifuge rotate if a particle 8.0 cm from the axis of rotation is to experience an acceleration of 100,000 g’s?

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Textbook Question

A cyclist accelerates from rest at a rate of 1.00 m/s². How fast will a point at the top of the rim of the tire (diameter = 68.0 cm) be moving after 2.75 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest—see Fig. 10–69.] <IMAGE>

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Textbook Question

On a 12.0-cm-diameter audio compact disc (CD), digital bits of information are encoded sequentially along an outward spiraling path. The spiral starts at radius R₁ = 2.5 cm and winds its way out to radius R₂ = 5.8 cm . To read the digital information, a CD player rotates the CD so that the player’s readout laser scans along the spiral’s sequence of bits at a constant linear speed of 1.25 m/s. Thus the player must accurately adjust the rotational frequency ƒ of the CD as the laser moves outward. Determine the values for ƒ (in units of rpm) when the laser is located at R₁ and when it is at R₂ .

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Textbook Question

(III) A hammer thrower accelerates the hammer of mass 7.30 kg (Fig. 10–64) from rest within four full turns (revolutions) and releases it at a speed of 26.5 m/s. Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate

(b) the (linear) tangential acceleration, (Ignore gravity)

<IMAGE>

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(d) the net force being exerted on the hammer by the athlete just before release, (Ignore gravity)

<IMAGE>

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Textbook Question

(e) the angle of this force with respect to the radius of the circular motion. (Ignore gravity)

<IMAGE>

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Textbook Question

(II) In traveling to the Moon, astronauts aboard the Apollo spacecraft put the spacecraft into a slow rotation to distribute the Sun’s energy evenly (so one side would not become too hot). At the start of their trip, they accelerated for 12 minutes from no rotation to 1.0 revolution per minute which they then maintained. Think of the spacecraft as a cylinder with a diameter of 8.5 m rotating about its cylindrical axis. Determine

(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 6.0 min after it started this acceleration.

100

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Textbook Question

The time-dependent position of a point object which moves counterclockwise along the circumference of a circle (radius R) in the xy plane with constant speed υ is given by

r→ = î R cos ωt + ĵ R sin ωt

where the constant ω = v/R. Determine the velocity v→ and angular velocity ω→ of this object and then show that these three vectors obey the relation v→ = ω→ x r→ .

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Textbook Question

(II) The platter of the hard drive of a computer rotates at 7200 rpm (rpm = revolutions per minute = rev/min)

(b) If the reading head of the drive is located 3.00 cm from the rotation axis, what is the linear speed of the point on the platter just below it?

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Textbook Question

(II) The platter of the hard drive of a computer rotates at 7200 rpm (rpm = revolutions per minute = rev/min)

(c) If a single bit requires 0.50 μm of length along the direction of motion, how many bits per second can the writing head write when it is 3.00 cm from the axis?

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Textbook Question

(II) A 61-cm-diameter wheel accelerates uniformly about its center from 120 rpm to 280 rpm in 5.0 s. Determine

(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating.

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