12. Rotational Kinematics

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.

(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?

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>

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 .

(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)

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(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

(c) the centripetal acceleration just before release, (Ignore gravity)

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(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

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

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(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

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

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(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.

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→ .

(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?

(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?

(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.