EXERCISES And PROBLEMS
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The Rotational Variables
IE. During a time interval 1 the flywheel of a generator turns through the angle () = at + hl3 – CI’, where a, b. and care constants. Write expressions for the wheel’s (a) angular velocity and (b) angular acceleration.
2E. What is the angular speed of (a) the second hand. (b) the minute hand, and (c) the hour hand of a smoothly running analog watch? Answer in radians per second.
3E. Our Sun is 2.3 X 104 Iy (light-years) from the center of our Milky Way galaxy and is moving in a circle around that center at a speed of 250 km/s. (a) How long does it take the Sun to make one revolution about the galactic center? (b) How many revolutions has the Sun completed since it was formed about 4.5 x 109 years ago?
4E. The angular position of a point on the rim of a rotating wheel is given by () = 4.01 – 3.012 + 1\ where () is in radians and I is in seconds. What are the angular velocities at (a) 1 = 2.0 s and (b) 1 = 4.0 s? (c) What is the average angular acceleration for the time interval that begins at t = 2.0 s and ends at 1 = 4.0 s? What are the instantaneous angular accelerations at (d) the beginning and (e) the end of this time interval?
5E. The angular position of a point on a rotating wheel is given by 8 = 2 + 4t2 + 2t3, where 8 is in radians and 1 is in seconds. At 1 = 0, what are (a) the point’s angular position and (b) its angular velocity? (c) What is its angular velocity at I = 4.0 s? (d) Calculate its angular acceleration at 1= 2.0 S. (e) Is its angular acceleration constant?
6P. The wheel has eight equally spaced spokes and a radius of 30 ern. It is mounted on a fixed axle and is spinning.
7P. A diver makes 2.5 revolutions on the way from a IQ-m-high platform to the water. Assuming zero initial vertical velocity, find the diver’s average angular velocity during a dive.
Rotation with Constant Angular Acceleration
8E. The angular speed of an automobile engine is increased at a constant rate from 1200 rev/min to 3000 rev/min in 12 S. (a) What is its angular acceleration in revolutions per minute-squared? (b) How many revolutions does the engine make during this 12 s interval?
9E. A record turntable rotating at 3~ rev/min slows down and stops in 30 s after the motor is turned off. (a) Find its (constant) angular acceleration in revolutions per minute-squared. (b) How many revolutions does it make in this time?
I0E. A disk, initially rotating at 120 rad/s, is slowed down with a constant angular acceleration of magnitude 4.0 rad/s2• (a) How much time does the disk take to stop? (b) Through what angle does the disk rotate during that time?
l1E. A heavy flywheel rotating on its central axis is slowing down .because of friction in its bearings. At the end of the first minute of slowing, its angular speed is 0.90 of its initial angular speed of 250 rev/min. Assuming a constant angular acceleration, find its
angular speed at the end of the second minute.
12E. Starting from rest, a disk rotates about its central axis with constant angular acceleration. In 5.0 s, it rotates 25 rad. During that time, what are the magnitudes of (a) the angular acceleration and (b) the average angular velocity? (c) What is the instantaneous angular velocity of the disk at the end of the 5.0 s? (d) With the angular acceleration unchanged, through what additional angle will the disk turn during the next 5.0 s?
13P. A wheel has a constant angular acceleration of3.0 rad/s2• During a certain 4.0 s interval, it turns through an angle of 120 rad.
14P. A wheel, starting from rest, rotates with’ a constant angular acceleration of 2.00 rad/s2. During a certain 3.00 s interval, it turns through 90.0 rad. (a) How long is the wheel turning before the start of the 3.00 s interval? (b) What is the angular velocity of the wheel at the start of the 3.00 s interval?
15P. At t = 0, a flywheel has an angular velocity of 4.7 rad/s, an angular acceleration of -0.25 rad/s2, and a reference line at lip = O. (a) Through what maximum angle will the reference line turn in the positive direction? At what times t will the reference line be at (b) II = !lImax and (c) II = -10.5 rad (consider both positive and negative values of t)? (d) Graph II versus t, and indicate the answers to (a), (b), and (c) on the graph.
16P. A disk rotates about its central axis starting from rest and accelerates with constau; angular acceleration. At one time it is rotating at 10 rev/s; 60 revolutions later, its angular speed is 15 rev/so Calculate (a) the angular acceleration, (b) the time required to complete the 60 revolutions, (c) the time required to reach the 10 rev/s angular speed, and (d) the number of revolutions from rest until the time the disk reaches.the 10 angular speed.
17P. A flywheel turns through 40 rev as it slows from an angular speed of 1.5 rad/s to a stop. (a) Assuming a constant angular acceleration, find the time lor H to come to rest. (b) What is its angular acceleration? (c) How much time is required for it to complete the
first 20 of the 40 revolutions?
18P. A wheel rotating about a fixed axis through its center has a constant angular acceleration , In a certain 4.0 s interval the wheel turns through an angle of 80 rad. (a) What is the angular velocity of the wheel at the start of the 4.0 s interval? (b) Assuming that the wheel starts from rest, how long is it in motion at the start of the 4.0 s interval.
Relating the linear and Angular Variables
19E. What is the linear acceleration of a point on the rim of a 30-em-diameter record rotating at a constant angular speed of 33~ rev/nun?
20E. A vinyl record on a turntable rotates at 3~ rev/min. (a) What is its angular speed in radians per second? Wh~t is the linear speed of a point on the record at the needle’ when the needle is (b) 15 cm and (c) 7.4 ern from the turntable axis?
21E. What is the angular speed of a car traveling at 50 km/h and rounding a circular turn of radius 110 n;?
22E. A flywheel with a diameter of 1.20 m is rotating at an angular speed of 200 rev/min. (a) What is the angular speed of the flywheel in radians per second? (b) What is the linear speed of a point on the rim of the flywheel? (c) What constant angular acceleration (in revolutions per minute-squared) will increase the wheel’s angular speed to 1000 rev/ruin in 60 s? (d) How many revolutions does the wheel make during that 60 s?
23E. An astronaut is being tested in a centrifuge. The centrifuge has a radius of 10 m and, in starting, rotates according to 0 = 0.30t2, where t is in seconds is in radians. When t = 5.0 s, what are the magnitudes of the astronaut’s.
24E. What are the magnitudes of (a) the angular velocity, (b) the radial acceleration, and (c) the tangential acceleration of a spaceship taking a circular turn ‘Of radius 3220 km at a speed of 29000 km/h?
2SP. An early method of measuring the speed of light makes use of a rotating slotted wheel. A beam of light passes through a slot at the outside edge of the wheel, as in Fig. 11-30, travels to a distant mirror, and returns to the wheel just in time to pass through the
next slot in the wheel. One such slotted wheel has a radius of 5.0 cm and 500 slots .
26P. The flywheel of a steam engine runs with a constant angular velocity of 150 rev/min. When steam is shut off, the friction of the bearings and of the air stops the wheel in 2.2 h. (a) What is the constant angular acceleration, in revolutions per minute squared, of the wheel during the slowdown? (b) How many rotations does the wheel make before stopping? (c) At the instant the flywheel is turning at 75 rev/min. what is the tangential component of the linear acceleration of a flywheel particle that is 50 em from the axis of rotation? (d) What is the magnitude of the net linear acceleration
of the particle in (c)?
27P. (a) What is the angular speed w about the polar axis of a point on Earth’s surface at a latitude of 40° N? (Earth rotates about that axis.) (b) What is the linear speed of the point? What are (c) w and (d) for a point at the equator?
28P. A gyroscope flywheel of radius 2.83 em is accelerated from rest at 14.2 rad/s? until its angular speed is 2760 rev/min. (a) What is the tangential acceleration of a point on the rim of the flywheel during this spin-up process?
28P. An object rotates about a fixed axis, and the angular position of a reference line on the object is given by (}= 0.40e21, where is in radians and t is in seconds. Consider a point on the object that is 4.0 ern from the axis of rotation.
29P A pulsar is a rapidly rotating neutron star that emits a radio beam like a lighthouse emits a light beam. We receive a radio pulse for each rotation of the star.
30P. A record turntable is rotating at 331 rev/min. A watermelon seed is on the turntable 6.0 ern from the axis of rotation.