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Unformatted text preview: Angular velocity and angular acceleration dθ ds i CHAPTER 9 r
ri ROTATION θi • Angular velocity and angular acceleration
o equations of rotational motion The arc length moved by the ith element in a rotating
• Torque and Moment of Inertia
o Newton’s 2nd Law for rotation
• Determination of the Moment of Inertia
• Rotational kinetic energy
• Rolling objects (with no slip) rigid, non-deformable disk is: dsi = ri dθ
where dθ is in radians. The angular velocity of the
rotating disk is defined as:
dt and so the linear velocity of the ith element (in the
direction of the tangent) is:
vi = dsi
= riω .
dt 1 What’s the relationship between the
angular and linear velocity of two points
on the same disk? If the angular velocity changes
there is angular acceleration ... dθ
ri θi r
a it r
a ir r
a ir ⊥ a it The angular acceleration of the disk is:
α= RMM02VD2.mov dω d dθ d 2θ
dt dt dt dt 2 and the tangential acceleration of the ith element is:
a it = i = ri
= riα .
dt The angular velocity ω is the same for all points on the But, because the ith element is traveling in a circle, it disk but the linear (or tangential) velocity is not. Look, experiences a radial (centripetal) acceleration: v = rω , so, since r2 > r1, then v 2 > v1. v2
a ir ( = a ic ) = i = riω 2 .
The resultant linear acceleration is a i = a ir 2 + a it2 . 2 CONNECTION BETWEEN LINEAR
AND ROTATIONAL MOTION Angular velocity ( ω ) (vector) Dimension: ω ⇒ Linear motion (Check: v i = riω ⇒ [L] 1 [L ]
[ T] [ T] Units: rad/s
Angular acceleration ( α ) (vector) 1
Dimension: α ⇒
[ T]2 Rotational motion a ⇒ constant 1
[ T] α ⇒ constant v = v o + at ω = ω o + αt 1
( x − x o ) = v o t + at 2
( θ − θ o ) = ω o t + αt 2
2 v 2 = v o2 + 2a ( x − x o ) ω 2 = ω o2 + 2α(θ − θo ) You see, they’re
very similar Units: rad/s2 3 r = 0.12 m: ω o = 0 : θo = 0 : t = 5s
r α = 3.00 rad/s2.
ω = ? : θ = ?: a t = ?: a c = ? (a) ω = ω o + αt = 3.00 × 5 = 15.0 rad/s . Question 1: A disk of radius 12cm, initially at rest,
begins rotating about its axis with a constant angular
2 acceleration of 3.00 rad/s . After 5s, what are (a) the
angular velocity of the disk, and (b) the tangential and
centripetal accelerations of a point on the perimeter of
the disk? (c) How many revolutions were made by the
disk in those 5s? (b) v i = riω = 0.12 × 15.0 = 1.80 m/s (linear).
• tangential acceleration: a t = riα
= 0.12 × 3.00 = 0.36 m/s2. • centripetal acceleration: a c = riω 2
= 0.12 × 15.02 = 27.0 m/s2. (Check ... a c = v 2 1.802
= 27.0 m/s2 .)
(c) (θ − θo ) = ω o t + αt 2 = × 3.00 × 52 = 37.5rad ,
= 5.97 rev .
2π 4 In many applications a belt or chain is pulled from or
wound onto a pulley or gear wheel ...
at DISCUSSION PROBLEM [9.1]: R As the string (chain or belt) is removed (or added), its
instantaneous velocity is the same as the tangential
velocity at the rim of the wheel, providing there is no
i.e., v t = Rω .
When you rewind an audio or video tape why does it Also, under the same conditions, the instantaneous wind up fastest when it has almost completed its task, acceleration of the string is the same as the tangential i.e., when the take-up spool is almost full? acceleration at the rim of the wheel:
= Rα .
i.e., a t = t = R
dt 5 As we saw in chapter 4 As we have seen, the magnitude of the torque is: force ⇒ change in motion,
the same is true for rotational motion. τ = lF = ( r sin θ)F . The tendancy for a force to cause rotation of a rigid body
is called torque ... consider a mass m attached to a The following demonstrations show the properties of this massless rigid rod that rotates around an axis O. A force expression. F will cause the mass to rotate. r
O l r
l = r sin θ The magnitude of the torque due to a force F on the
τ = lF = ( r sin θ)F , where l is called the lever arm. RMM05VD1.MOV The lever arm is the perpendicular distance from the
axis of rotation (O) to the line of action of the force. 6 Dimension: τ ⇒ [L] [M][L] [M][L]2
[ T]2 (vector) Ft = F sin θ Unit: N ⋅ m
r Form the radial and tangential components of the
Ft = F sin θ r
m Fr = F cos θ l Newton’s 2nd Law tells us that the tangential component r
m O r
F of the force Ft produces a tangential acceleration a t ,
Fr = F cos θ l i.e., Ft = ma t .
∴ τ = rFt = mra t . But, from Chapter 8, the tangential (linear) acceleration
is related to the angular acceleration α, viz: a t = rα . Then τ = lF = ( r sin θ)F = r (F sin θ) ∴ τ = mra t = mr 2α . i.e., τ = rFt .
Note: the radial component Fr , which passes through A rigid object that rotates about a fixed axis is really just the axis of rotation, does not produce a torque and a collection of individual elements of mass ( ∆mi ) that therefore it does not produce rotation; only the each move in a circular path of radius ri, where ri is tangential component Ft produces a torque that results measured from the axis of rotation in rotation. 7 From above, for each mass element, we have τi = ( ∆mi ) ri2α ,
ri where τi is the net
torque on the ith ∆mi O element. Summing over
all elements, the total net
torque on the object is: ( ) τ net = ∑i τi = ∑i ( ∆mi ) ri2α = ∑i ( ∆mi ) ri2 α
RMM05VD2.MOV = Iα , where we call I = ∑i ( ∆mi ) ri2 the MOMENT OF
INERTIA. This is Newton’s 2nd Law for rotation, i.e.,
τ net = Iα . Same torques ( τ1 = τ2 ).
Different moments of inertia ( I1 > I2).
But τ = Iα
∴ α1 < α 2 “ A net external torque acting on a body produces
an angular acceleration, α , of that body given
by Iα , where I is the moment of inertia.”
(viz: Fnet = ∑i Fi = ma .) 8 Moment of inertia ... so
what’s that all about? Dimension: I ⇒ [M][L]2 (scalar) 2 Units: kg ⋅ m Every object has a moment of inertia about an axis of Question 2: Find the moment of inertia of a uniform thin
rod of length l and mass M rotating about an axis
perpendicular to the rod and through its center.
y rotation. Its value depends on how the mass is
x distributed around that axis. For a collection of n
objects, the moment of inertia about the rotation axis is: l I = ∑ n I n = ∑ n mn rn2.
For a “continuous” object: I = Limit ∑i ( ∆mi )ri2 = ∫ r 2dm,
∆mi →0 where m is a function of r. 9 y  Show for yourselves that the moment of inertia of a dx x x − l2 l rod of mass M and length l about one end is 2 y Since the rod is a “continuous” object, the moment of
inertia is I = ∫ r 2dm = x= l 0 1
I = Ml2.
3 2 the element of length dx is () ∫ x= − l 2 l3 distance from one end is
l ( l) l
( − l3 ) 2 x3 2
l3 − l y () x 2 M dx = M 1
= Ml −
= Ml2 . 24 l
−2 ()  Show for yourselves that the moment of inertia of a
rod of mass M and length l about an axis one-third the dm = M l dx . ∴I = l 2 x= − l x x 2
∫ x dm. The mass per unit length of the rod is M l , so the mass of x= l 2 dx dx 2 − l3 0 x x 2l 3 1
I = Ml2.
9 10 The moment of inertia is given by
I = ∫ r 2dm. r1 Consider a ring of
radius r and width dr. dr
r Question 3: Find the moment of inertia of the circular If the mass of the object shown below, rotating about an axis perpendicular
to the plane and through its center. The mass is 1.50kg. object is M, the mass
of the ring is dm = M 2πrdr
π r22 − r12 ( ) where r1 and r2 are the inner and outer radii of the object
and M its mass. Then
I = ∫ r 2M 10cm r1 20cm 2πrdr
2M r2 3
∫ r dr
π r22 − r12
r2 − r12 r1 ( )( ) r 2
2M r 4 M
r 4 − r14
r2 − r1 r
1 ( = )( ( ) ) M
r22 − r12 r22 + r12 = M r22 + r12 .
2 r22 − r12 ( )( )( ) ( ) 11 Sure, but, what’s the significance of I? 1
∴ I = M r22 + r12 = × 1.50 × 0.202 + 0.102
2 ( ( ) ) = 3.75 × 10 −2 kg ⋅ m2 . 1
 If r1 = 0 , then Idisk = MR 2 , where R is the radius of
the disk ( = r2 ) . R Remember, from chapter 4 ...
Mass ⇒ a measure of resistance to a change in
linear motion, e.g., how difficult it is to start or
stop linear motion.
Moment of Inertia ⇒ a measure of resistance to a
change in rotational motion, i.e., how difficult it is  For a thin hoop, r1 ≈ r2 = R , then I hoop = MR 2. to start or stop rotational motion. R Note that providing the thicknesses of the disks are
uniform, the moments of inertia do not depend on
Different moments of inertia 12 Values of the moment of inertia for “simple” shapes ...
Thin rod Slab 1
I = ML2
I = M(a 2 + b 2 )
12 b b L L a a
I = Ma 2
3 Thin rod
2 I = ML
3 Question 4: A 1m ruler has a mass of 0.25kg. A 5kg Hollow cylinder
I = M( R 2 + r 2 ) Hollow sphere
I = MR 2 2 3 mass is attached to the 100cm end of the rule. What is
its moment of inertia about the 0cm end? Thin-walled cylinder
2 I = MR Solid cylinder
2 I = MR
2 Solid sphere
2 I = MR
5 13 Parallel axis theorem:
O 5kg b Usually, the moment of inertia D a is given for an axis that passes O′ Since a >> b we assume the meter rule is a slab with
I = Ma 2 about the O − O′ rotation axis.
I ruler → Ma 2 = × 0.25 × 12 = 0.083kg ⋅ m2 .
The moment of inertia of the 5kg mass about O − O′ is
2 2 2 I mass = ma = 5 × 1 = 5kg ⋅ m .
Thus, the total moment of inertia of the ruler and mass is • cm through the center of mass
(cm) of the object. The
moment of inertia about a general (parallel) axis is given by:
I = Icm + MD2 where Icm is the moment of inertia about the center of
mass, M is the mass of the object and D is the distance
between the axes. 2 I ruler + I mass = 5.083kg ⋅ m .
Note: the two rotation axes must be parallel 14 y y
3kg 4kg 2m Question 5: Four masses at the corners of a square with
2m side length L = 2 m are conncted by massless rods. The
masses are m1 = m3 = 3kg and m2 = m4 = 4 kg. Find (a)
the moment of inertia about the z-axis, (b) the moment 4kg •cm D 3kg
x z 2m z x of inertia about an axis that is perpendicular to the plane
of the ensemble and passes through the center of mass of
the system, (c) the moment of inertia about the x-axis, (a) Moment of inertia about the z-axis: I z = ∑i mi ri2 = 3 × 22 + 4 × (2 2 )2 + 3 × 22 + 4 × 0 = 56 kg ⋅ m2. which passes through m3 and m4, and (d) the moment of
inertia about the y-axis, which passes through m1 and
m2. (b) By symmetry, the center of mass is at the center of
y the square.
L m1 ∴ Icm = ∑i mi ri2 m2 = 3 × ( 2 )2 + 4 × ( 2 )2 + 3 × ( 2 )2 + 4 × ( 2 )2 L = 28 kg ⋅ m2.
x • Check, using the parallel axis-theorem I z = Icm + MD2
∴ Icm = I z − MD2 = 56 − 14 × ( 2 )2 = 28 kg ⋅ m2 . 15 We saw in chapter 6 that a linearly moving object has
y 2m 2m 3kg 4kg 4kg an axis has rotational kinetic energy ... 2m 2m z translational kinetic energy ... an object rotating about y •cm If a rigid object is rotating 2m 3kg
x z (c) Moment of inertia about the x-axis: I x = ∑i mi ri2 = 3 × 22 + 4 × 22 = 28 kg ⋅ m2.
(d) Moment of inertia about the y-axis:
I y = ∑i mi ri2 = 4 × 22 + 3 × 22 = 28 kg ⋅ m2. x ω
O ri vi
∆mi with angular velocity ω ,
the kinetic energy of the ith
Ki = ( ∆mi ) v i2
= ( ∆mi )( ri2ω 2 ) , since v i = riω .
2 So, the total rotational kinetic energy is:
K = ∑i Ki = ∑i ( ∆mi )ri2ω 2
= Iω 2.
* K rot = Iω 2 is the analog of K trans = mv 2 .
2 16 A torque is required to rotate (or slow down) an object ...
but torque involves force ... r
F ds Ft r
O Fr γ + θ = 90o
Ft = F cos γ Question 6: An engine develops 400N ⋅ m of torque at
3700rev/min. What is the power developed by the When force is applied over a distance, work is done,
given by: engine? rr
dW = F • ds = F.ds cos γ = F cos γ .ds
= Ft .ds = Ft .rdθ = τ.dθ (J or N.m). Power is the rate at which the torque does work,
= τ = τω
i.e., P =
• dW = τ.dθ is the analog of dW = F.ds .
• P = τω is the analog of P = Fv . 17 Torque τ = 400N ⋅ m.
Angular velocity ω = 3700 rev/min
3700 × 2π
From earlier, power P = τω = 400 × 387.5 = 1.55 × 105 watts.
But 746 watts = 1HP 10N ⋅ m to the shaft of a grindstone. If the moment of inertia of the grindstone is 2 kg ⋅ m2 and the system starts
from rest, find (a) the rotational kinetic energy of the
grindstone after 8.0s, (b) the work done by the motor
during this time, and (c) the average power delivered by 5 ∴P = Question 7: An electric motor exerts a constant torque of 1.55 × 10
= 208HP .
746 the motor. 18 (a) To find the rotational kinetic energy we need to know
the angular velocity ω .
Since τ = Iα , α = = = 5.0 rad/s2.
The grindstone starts from rest so ω = αt = 5.0 × 8.0 = 40 rad/s.
∴ K = Iω 2 = × 2 × 402 = 1600J .
(b) There are two ways to determine the work done by
(i) By the work-kinetic energy theorem we would
expect the motor to have done 1600J of work.
(ii) We can use the expression W = τθ , but we need (c) The average power is the total work done divided by
the time interval, i.e.,
= 200 W .
Note: the expression P = τω we derived earlier is actually
the instantaneous power, as ω is not constant. The
instantaneous power increases linearly from zero at t = 0
to (10 × 40) = 400 W at t = 8.0s. to find θ , i.e., the angle through which the grindstone has
turned in 8.0s.
θ = αt 2 = × 5.0 × (8.0)2 = 160 rad
2 (25.46 rev). ∴ W = τθ = 10 × 160 = 1600J . 19 2v cm If a rigid object is suspended from an arbitrary point O
and is free to rotate about that point, it will turn until the v cm ω center of mass is vertically beneath the suspension point.
v=0 Consider a ball (cylinder or
wheel, etc) rolling on a surface
without slipping. y • The point in contact with the surface has zero
instantaneous velocity relative to the surface. rcm × cm
O xcm x Mg • The velocity of the cm is v cm ( = rω = v ),
• The velocity of a point at the top is 2 v cm ( = 2 v ).
Since the object has the same linear velocity as the cm, If the y-direction is vertical and the suspension point is
not at the center of mass, the object will experience a net i.e., v, the total kinetic energy is:
K tot = mv 2 + Iω 2
2 torque given by τ = Mgx cm,
where x cm is the x component of the center of mass.
Therefore, the object will rotate until x cm = 0, i.e., the
suspension point is direction above the center of mass. translational + rotational
where ω = v r . So, as a ball rolls down a hill ...
v, ω Gravitational potential energy ⇒
+ rotational energy 20 Consider rolling a sphere, cylinder and a ring down an
incline. Do they travel with the same velocity? h v For each object, conservation of energy gives:
mgh = mv 2 + Iω 2
and with no slip v = Rω , where R is the radius of the
object. After manipulation we find:
. 1+ I mR 2 ∴ For maximum velocity: I ⇒ smallest. RMA07VD2.MOV v2 = 2 gh 1+ I mR 2 2
For a solid sphere I = mR 2 .
, i.e., v =
1 + 25 Since this is independent of m and R, a bowling ball and
a pool ball would have the same speeds at the bottom of So, in a race between objects rolling down a slope, the an incline, despite their different masses and radii! order would be (1) sphere, (2) cylinder, (3) hoop, and is
completely independent of m and R! 21 Draw the free body diagram of all the forces acting on
the sphere. Use Newton’s 2nd Law along the incline: Mg sin 30o − f = Ma cm , N f R horizontal. Express the frictional force acting at the point
of contact with the surface in terms of M, R and g. where a cm is the acceleration
of the center of mass. To get Mg Question 8: A uniform sphere of mass M and radius R
rolls without slipping down an incline at 30o to the 30o O an expression for a cm, we take
torques about O, the center of the disk. τ = fR = Iα ,
where I is the moment of inertia of the sphere and α its
angular acceleration. (Note that the normal force N and
the weight force do not contribute to the torque as their 30 o lines of action pass through O.) With no slip a cm = Rα = fR 2
I Substituting for a cm in the first equation, we get Mg sin 30o − f = M fR 2
2MR 2 5 5
2 22 Mg
∴ f = Mg ⇒ f = 2
F You can show yourselves that if we put I = βMR 2, where
β = 1 for a hoop, β = 12 for a disk, etc., and the slope of
the incline is θ, then a general expression for the x R frictional force acting on an object, with no slip, is
Mg sin θ
1 + β −1 Question 9: A cue ball is struck by a horizontal cue a You can also show that since in terms of the radius R of the ball. a cm = fR 2
I distance x above the center of the ball. If the cue ball is
to roll without slipping, what is x? Express your answer You can assume that the frictional force of the table on
the ball is negligible compared with the applied force F. (from earlier) then
a cm = g sin θ
1+ β 23 F
x R • The force produce a torque about the center τ = Fx .
From earlier, if there is no-slip then v cm = Rω ,
= Rα .
i.e., a cm = cm = R
Using Newton’s 2nd Law
F = ma cm and τ = Iα = Fx .
∴ = a cm = Rα = R , i.e., x =
For a sphere I = mR 2 .
∴ x = R.
• x > R ⇒ top spin
• x < R ⇒ back spin
5 m2 30kg
20kg m1 2m Question 10: The system shown above is released from
rest, with the 30kg mass 2m above the ground. Modeling
the pulley as a uniform disk with a radius of 10cm and
mass 5kg, find (a) the speed of the 30kg mass just before
it strikes the ground, (b) the angular velocity of the
pulley at that instant, (c) the tensions in the two strings,
and (d) the time it takes for the 30kg block to reach the
ground. Assume the bearings in the pulley are
frictionless and there is no slip between the string and the
pulley. 24 • ω •
m2 30kg m1 2m m2 ω •
m2 30kg v 20kg m1 •
v v 2m 20kg m1 We have both translational and rotational motion. The
moment of inertia of the pulley is:
I = mR 2 = × 5 × 0.102 = 0.025kg ⋅ m2.
(a) Conservation of energy ... as m2 hits the ground
m2 gh = m1v 2 + m1gh + m2 v 2 + Iω 2 .
The angular velocity of the pulley ω = , so
30 × 9.81 × 2 = × 20 v 2 + 20 × 9.81 × 2
+ × 30 v 2 + × 0.025 ×
(0.10)2 (b) ω = (c) m2 v v 2.73
R 0.10 T1
a Draw the free-body a m2 g diagrams for the masses. What is the upward acceleration of m1? v 2 = 2ah . ∴ a = v 2 2.742
= 1.88 m/s2.
2h 2 × 2 Then T1 − m1g = m1a ,
i.e., T1 = m1 ( g + a ) = 234 N.
Also T2 − m2 g = m2 ( −a ),
i.e., T2 = m2 ( g − a ) = 238 N. i.e., 589 = 10 v 2 + 392 + 15v 2 + 1.3v 2
∴v = 197
= 2.74 m/s.
(d) ( y − y o ) = h = at 2. ∴ t =
2 25 ...
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