Ch 7 - Rotational Motion Gravitation - Slides Practice.pdf

Raymond A. Serway
Chris Vuille
Chapter 7
Rotational Motion and Gravitation

2
Rotational Motion
• The concept of rotational motion is central to understanding
the motions of a diverse range of phenomena, from a car
moving around a circular racetrack to clusters of galaxies
orbiting a common center.
• Rotational motion, when combined with Newton’s law of
universal gravitation and his laws of motion, can also explain
certain facts about space travel and satellite motion, such as
where to place a satellite so it will remain fixed in position
over the same spot on the Earth.
• The generalization of gravitational potential energy and
energy conservation offers an easy route to such results as
planetary escape speed.

3
The Radian
• The radian is a unit of angular measure.
• The radian can be defined as the arc length s along a circle
divided by the radius r.
• Converting from degrees to radians:
• Generally, angular quantities in physics must be expressed in
radians.
• Be sure to set your calculator to radian mode.
s
r
 =
]
rees
[deg
180
]
rad
[ 


=


4
Angular Displacement
The point P on a rotating
compact disc at t = 0.
As the disc rotates, P moves
through an arclength s.
• An object’s angular displacement, ΔƟ, is the difference in its final
and initial angles, .
• SI unit: radian (rad)

5
Angular Velocity

6
Tips and unit Conversions

7
Example 1
The rotor of a helicopter turns at an angular speed of 320.0 rev/min.(a) If the rotor has a
radius of 2.00 m, what arclength does the tip of the blade trace out in 5.00 minutes. (b) The
pilot opens the throttle, and the angular speed of the blade increases while rotating twenty-
six times in 3.6 s. Calculate the average angular speed during this time.

Rotational Motion and Gravitation 8
Angular Acceleration
• An object’s average angular acceleration during the time interval
Δt is the change in its angular velocity Δω divided by Δt:
• SI unit: radian per second squared (rad/s²)
• We take α to be positive when ω is increasing (counterclockwise
motion) and negative when ω is decreasing (clockwise motion).
• When a rigid object rotates about a fixed axis every portion of the
object has the same angular velocity and the same angular
acceleration.

9
Equations of Rotational Kinematics
The resulting equations of rotational kinematics, along with the corresponding
equations for linear motion.

10
Example 2
A wheel rotates with a constant angular acceleration of 3.50 rad/s². If the angular velocity of
the wheel is 2.00 rad/s at t = 0, (a) through what angle does the wheel rotate between t = 0
and t = 2.00 s? Give your answer in radians and in revolutions. (b) What is the angular
velocity of the wheel at t = 2.00 s?

Example 3
A rotating wheel requires 3.00 s to rotate 37.0 revolutions. Its angular velocity at the end of
the 3.00 s interval is 98.0 rad/s. What is the constant angular acceleration of the wheel?
Ɵ = number of revolutions x 2∏
Ɵ = 37 rev x 2∏ → Ɵ = 232.48 rad
232.48 = 0.5(ωi + 98) (3) → ωi=56.98 = 57.0 rad/s
98 = 57 + α (3) → α = 13.7 rad/s²

12
Relationships between Linear and Rotational Quantities
Tangential Speed - Tangential & Centripetal Acceleration

Relationships between Linear and Rotational Quantities
Centripetal Force & Total Acceleration

Example 4
A compact disc (CD) rotates from rest up to an angular velocity of 31.4 rad/s in a time of 892
ms. (a) What is the angular acceleration of the disc, assuming the angular acceleration is
uniform? (b) Through what angle does the disc turn while coming up to speed? (c) If the
radius of the disc is 4.45 cm, find the tangential velocity of a microbe riding on the rim of the
disc when t = 892 ms. (d) What is the magnitude of the tangential acceleration of the
microbe at the given time?

15
Example 5
In a compact disc player, as the read head moves out from the center of the disc, the angular speed of
the disc changes so that the linear(tangential) speed at the position of the head remains at a constant
value of 1.3 m/s. (a) Find the angular speed of a compact disc of radius 6.00 cm when the read head is at
r = 2.0 cm and again at r = 5.6 cm. (b) An old fashioned record player rotates at a constant angular speed,
so the linear speed of the record groove moving under the detector(stylus) changes. Find the linear
speed of a 0.750 rev/s(rps) record at points 2.0 cm and 5.6 cm from the center. (c) In both the CDs and
phonograph records, information is recorded in a continuous spiral track. Calculate the total length of
the track for a CD designed to play for 1.0 h

16
Practice 1
Centripetal Acceleration
A test car moves at a constant speed around a circular track. If the car is 18.2 m from the track’s center
and has a centripetal acceleration of 8.05 m/s², what is the car’s tangential speed?

17
Example 6
A race car accelerates uniformly from a speed of 40.0 m/s to a speed of 60.0 m/s in 5.00 s
while traveling counterclockwise around a circular track of radius 4.00 x 10² m. When the car
reaches a speed of 50.0 m/s, calculate (a) the magnitude of the car’s centripetal acceleration,
(b) the angular velocity, (c) the magnitude of the tangential acceleration, and (d) the
magnitude of the total acceleration.

18
Practice 2
Centripetal Force
A pilot is flying a small plane at 56.6 m/s in a circular path with radius 188.5 m. The centripetal force
needed to maintain the plane’s circular motion is 18.9 kN. What is the plane’s mass?

Problem Solving Strategy

20
Practice 3
Force of Friction (Fs = Fc)
(i) A 13.5 kN car traveling at 50 km/h rounds a curve of radius 200 m. Find (a)
the centripetal acceleration of the car. (b) The centripetal force. (c) the
minimum coefficient of static friction between the tires and the road that will
allow the car to round the curve safely. (mass = weight/9.8)
(a) ac = v²/r = (50x1000/3600)²/200 = 0.965 m/s²
(b) Fc = m.ac = (13.5x1000/9.8) x 0.965 = 1329.3 N
(c) Fs = Fc =1329.3 N and Fn = Fg = 13500 N
→ µ = Fs/Fn = 1329.3/13500 = 0.985
(ii) A 2x10³ kg car rounds a circular turn of radius 20.0 m. If the road is flat
and the coefficient of static friction between the tires and the road is 0.70,
how fast can the car go without skidding?

Newton’s Law of Universal Gravitation

Practice 4 – Newton’s Law of Universal Gravitation
(i) Find the distance between a 0.30 kg billiard ball
and a 0.40 kg billiard ball if the magnitude of the
gravitational force between them is 8.92x10-11 N.
(ii) A 600 kg satellite is in a circular orbit about
Earth (5.972 x 10²⁴ kg) at a height above Earth
equal to Earth's mean radius (6371 km). Find:
(a) The gravitational force acting on it,
(b) The satellite's orbital linear and rotational
speeds.
(iii)
(b) What is the electron orbital linear speed?
(c) What is the electron angular speed?

Newtonian Gravitation
• Gauss’ Law: The gravitational force
exerted by a uniform sphere on a
particle outside the sphere is the same
as the force exerted if the entire mass
of the sphere were concentrated at its
center.
• The free-fall acceleration g varies
considerably with altitude above the
Earth.

Measurement of the Gravitational Constant
• The gravitational constant G was first measured in an important
experiment by Henry Cavendish in 1798.
• The smaller spheres of mass m are attracted to the large spheres of
mass M, and the rod rotates through a small angle. A light beam
reflected from a mirror on the rotating apparatus measures the
angle of rotation.

Problems

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Problems

Ch 7 - Rotational Motion Gravitation - Slides Practice.pdf

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