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09 homework FRENNEA, KYLE Due: Mar 28 2007, 4:00 am Solution: Horizontally Question 1 part 1 of 1 10 points Two ice skaters approach each other at right angles. Skater A has a mass of 72.5 kg and travels in the +x direction at 2.88 m/s. Skater B has a mass of 62.5 kg and is moving in the +y direction at 0.982 m/s. They collide and cling together. Find the final speed of the couple. Correct answer: 1.6121 m/s (tolerance 1 %). Explanation: From conservation of momentum p = 0 mA vA^ + mB vB^ = (mA + mB ) vf i Therefore vf = (mA vA )2 + (mB vB )2 mA + mB (208.8 kg m/s)2 + (61.375 kg m/s)2 = 72.5 kg + 62.5 kg = 1.6121 m/s Question 2 part 1 of 2 10 points
before m1 v1 m1 after v3 v4 v2 m2 m2
1
m1 v1 + m2 (0) = m1 v3 cos + m2 v4 cos , so that m2 v4 cos = m1 v1 - m1 v3 cos (1) = (0.6 kg) (8.2 m/s) - (0.6 kg) (6.4 m/s) cos 67 = 3.41959 kg m/s . Vertically m1 (0) + m2 v2 = m1 v3 sin + m2 v4 sin , so that m2 v4 sin = m2 v2 - m1 v3 sin (2) = (1.6 kg) (5.6 m/s) - (0.6 kg) (6.4 m/s) sin 67 = 5.42526 kg m/s . Thus tan = m2 v4 sin m2 v4 cos (5.42526 kg m/s) = (3.41959 kg m/s) = 1.58652 ,
and = arctan(1.58652) = 57.7764 . Question 3 part 2 of 2 10 points With what speed does the can move immediately after the collision? Correct answer: 4.00815 m/s (tolerance 1 %). Explanation: Using equation (1) above, v4 = m1 v1 - m1 v3 cos m2 cos (0.6 kg) (8.2 m/s) = (1.6 kg) cos 57.7764 (0.6 kg) (6.4 m/s) cos(67 ) - (1.6 kg) cos(57.7764 ) = 4.00815 m/s
An m2 = 1.6 kg can of soup is thrown upward with a velocity of v2 = 5.6 m/s. It is immediately struck from the side by an m1 = 0.6 kg rock traveling at v1 = 8.2 m/s. The rock ricochets off at an angle of = 67 with a velocity of v3 = 6.4 m/s. What is the angle of the can's motion after the collision? Correct answer: 57.7764 (tolerance 1 %). Explanation: Basic Concepts: Conservation of Momentum pbef ore = paf ter .
homework 09 FRENNEA, KYLE Due: Mar 28 2007, 4:00 am or using equation (2) above, m2 v2 - m1 v3 sin m2 sin (5.6 m/s) = sin(57.7764 ) (0.6 kg) (6.4 m/s) sin(67 ) - (1.6 kg) sin(57.7764 ) = 4.00815 m/s . p2 N m2 v2 p1 m1 v1 pf = mf vf
2
v4 =
Basic Concepts: Momentum Conserva1 p = mv. tion, K = mv 2 , 2 During the collision, the total momentum of the two car system will be conserved pf = p1 + p2 = m1 v1 + m2 v2 pf x ^ + pf y ^ = m1 v1 ^ + m2 v2 ^ . i i Looking at the x and y components of momentum, pf x = m2 v2 = (3651.6 kg) (180 km/h) 103 m 1h km 3600 s = 182580 kg m/s , and pf y = m1 v1 = (2040 kg) (219.6 km/h) 1h 103 m km 3600 s = 124440 kg m/s . Since we are asked to find the angle from the y-axis instead of the x, tan = pf x pf y 182580 kg m/s = 124440 kg m/s = 1.46721 182580 kg m/s = arctan 124440 kg m/s = 55.7231 . Question 5 part 2 of 2 10 points
Question 4 part 1 of 2 10 points A 2040 kg car skidding due north on a level frictionless icy road at 219.6 km/h collides with a 3651.6 kg car skidding due east at 180 km/h in such a way that the two cars stick together. 180 km/h N 3651.6 kg 219.6 km/h 2040 kg vf
At what angle (-180 +180 ) East of North do the two coupled cars skid off at? Correct answer: 55.7231 (tolerance 1 %). Explanation:
Let : m1 v1 m2 v2
= 2040 kg , = 219.6 km/h , = 3651.6 kg , and = 180 km/h .
homework 09 FRENNEA, KYLE Due: Mar 28 2007, 4:00 am How much kinetic energy is lost in the collision? Correct answer: 4.07108 106 J (tolerance 1 %). Explanation: To find the energy lost in the collision, we need to know the total kinetic energy of the system before and after the collision. (There is no change in potential energy since the cars remain on level ground.) The initial and final energies are given by Ei = Ki = = 1 1 2 2 m1 v1 + m2 v2 2 2 Question 6 part 1 of 2 10 points
3
A ball moving at 18 m/s makes an offcenter elastic collision with another ball of equal mass that is initially at rest. The incoming ball is deflected at an angle of 43 from its original direction of motion. Find the speed of the first ball after the collision. Correct answer: 13.1644 m/s (tolerance 1 %). Explanation:
1 (2040 kg) (61 m/s)2 2 1 + (3651.6 kg) (50 m/s)2 2 = 8.35992 106 J , and Ef = Kf 1 2 = (m1 + m2 ) vf . 2 To find vf , we need to go back to the momentum equation; pf , m1 + m2 or in component form, pf x vf x = , m1 + m2 pf y . vf y = m1 + m2 vf =
Let : v = 18 m/s = 43 .
and
Because of the elastic collision, and the fact that the masses are equal, the angle of deflection for the recoiling ball is 47 . Applying conservation of momentum parallel to the original line of motion
and
pxi = pxf m v = m v1 cos 1 + m v2 cos 2 v2 cos 2 = v - v1 cos 1 .
(1)
Now that we know the components of vf , we can find its magnitude
2 2 2 vf = vf x + vf y .
Applying conservation of momentum perpendicular to the original line of motion, pyi = pyf 0 = m v1 sin 1 - m v2 sin 2 v2 sin 2 = v1 sin 1 . Dividing Eq. 2 by Eq. 1, v2 sin 2 v1 sin 1 = v2 cos 2 v - v1 cos 1 tan 2 (v - v1 cos 1 ) = v1 sin 1 v1 (sin 1 + tan 2 cos 1 ) = v tan 2
We can substitute this into the equation for the final kinetic energy to get Ef = 1 2 m1 + m2 = 4.28884 106 J . p2 x f + p2 y f
(2)
The energy lost is then Elost = -E = Ei - Ef
= (8.35992 106 J) - (4.28884 106 J) = 4.07108 106 J .
homework 09 FRENNEA, KYLE Due: Mar 28 2007, 4:00 am
4
v1 =
v tan 2 sin 1 + tan 2 cos 1 (18 m/s) tan 47 = sin 43 + tan 47 cos 43
If the car is moving at a constant speed, there is no tangential acceleration, thus the acceleration is purely radial, ar = v2 . R
= 13.1644 m/s . Question 7 part 2 of 2 10 points Find the speed of the second ball. Correct answer: 12.276 m/s (tolerance 1 %). Explanation: The magnitude of the velocity is v2 = v1 sin 1 sin 2 (13.1644 m/s) sin 43 = sin 47 = 12.276 m/s . Question 10 part 1 of 3 10 points The figure below represents, at a given instant, the total acceleration of a particle moving clockwise in a circle of radius 1.63 m. The magnitude of the total acceleration of the particle at the given instant is 10.5 m/s2 ; the angle between the vector of total acceleration and the radius-vector of the particle is 30 . v
1. 63
m
Question 8 part 1 of 2 10 points A racing car travels on a circular track of radius 163 m. The car moves with a constant linear speed of 20.9 m/s. Find its angular speed. Correct answer: 0.128221 rad/s (tolerance 1 %). Explanation: The linear speed v and the angular speed are related by, v = R v = . R Question 9 part 2 of 2 10 points Find the magnitude of its acceleration. Correct answer: 2.67982 m/s2 (tolerance 1 %). Explanation:
30
At this instant of time, find the centripetal acceleration of the particle. Correct answer: 9.09327 m/s2 (tolerance 1 %). Explanation: The radial (i.e., centripetal) acceleration ar of the particle is ar = a cos = (10.5 m/s2 ) cos (30 ) = 9.09327 m/s2 .
Question 11 part 2 of 3 10 points Find the speed of the particle at the same moment. Correct answer: 3.84994 m/s (tolerance 1 %). Explanation:
10.5 m/s2
homework 09 FRENNEA, KYLE Due: Mar 28 2007, 4:00 am From the formula for the radial (i.e., centripetal) acceleration v2 ar = r we obtain that the speed of the particle is v = r ar = (1.63 m) (9.09327 m/s2 ) Question 14 part 1 of 1 10 points = 3.84994 m/s . Question 12 part 3 of 3 10 points Find particle's tangential acceleration at the same moment. Correct answer: 5.25 m/s2 (tolerance 1 %). Explanation: From the formula Solution: t = avg 3.1 rad = 2.2 rad/s = 1.40909 s
5
A woman passes through a revolving door with a tangential speed of 1.2 m/s. If she is 0.62 m from the center of the door, what is the door's angular speed? Correct answer: 1.93548 rad/s (tolerance 1 %). Explanation: Basic Concept: vt = r
a2 = a2 + a2 t r we obtain at = = a2 - a2 r (10.5 m/s2 )2 - (9.09327 m/s2 )2 Given: vt = 1.2 m/s r = 0.62 m Solution: = vt r 1.2 m/s = 0.62 m = 1.93548 rad/s
= 5.25 m/s2 .
Question 13 part 1 of 1 10 points A girl ties a toy airplane to the end of a string and swings it around her head. The plane's average angular speed is 2.2 rad/s. In what time interval will the plane move through an angular displacement of 3.1 rad? Correct answer: 1.40909 s (tolerance 1 %). Explanation: Basic Concept:
Question 15 part 1 of 2 10 points t You want to design a large, permanent space station so that no artificial gravity is necessary. You decide to shape it like a large coffee can of radius 164 m and rotate it about its central axis. The acceleration of gravity is 9.8 m/s2 .
avg = Given:
avg = 2.2 rad/s = 3.1 rad
homework 09 FRENNEA, KYLE Due: Mar 28 2007, 4:00 am Thus the effective gravity the jogger feels is
6
radius 164 m
ag =
2 vef f
r [r + vm ]2 = r [(164 m) (0.244451 rad/s) + (2.7 m/s)]2 = (164 m) = 11.1645 m/s2 . Question 17 part 1 of 1 10 points
What rotational speed would be required to simulate gravity? Correct answer: 0.244451 rad/s (tolerance 1 %). Explanation: Basic Concepts The centripetal acceleration is the simulated gravity. ac = g = so that = = g r (9.8 m/s2 ) (164 m) v2 (r )2 = = r 2 r r
A girl throws a long stick with a length of 0.46 m and a mass of 0.2 kg into the air in such a way that the center of mass rises vertically. At the moment it leaves her hand, the stick is horizontal and the speed of the end of the stick nearest to her is zero. When the center of mass of the stick reaches its highest point, the stick is horizontal, and it has made 18 complete revolutions. Assume: The stick's cross sectional area and mass is uniform. The acceleration of gravity is 9.8 m/s2 . vcm0 0.2 kg 0.23 m 0.46 m vend0
= 0.244451 rad/s .
Question 16 part 2 of 2 10 points If an astronaut jogged in the direction of the rotation at 2.7 m/s , what is the simulated gravitational acceleration the astronaut would feel? Correct answer: 11.1645 m/s2 (tolerance 1 %). Explanation: The linear speed of the rim is vrim = r , so that his effective speed is vef f = vrim + vm = r + vm . How long did it take for the center of mass to reach its highest point? Correct answer: 1.62921 s (tolerance 1 %). Explanation: From kinematics
v0 = g t From definition of angular velocity v0 = r = From kinematics =t = . t . 2
(1)
(2)
(3)
homework 09 FRENNEA, KYLE Due: Mar 28 2007, 4:00 am Substituting from Eq. 3 into Eq. 2 gives v0 = . 2 t = gt. 2 t Since, = n (2 ) , we have t= = n g (6) (4) Question 20 part 1 of 2 10 points
7
Substituting v0 from Eq. 1 into Eq. 4 gives (5)
(18) (0.46 m) (9.8 m/s2 ) = 1.62921 s .
A small turtle, appropriately named "Dizzy", is placed on a horizontal, rotating turntable at a distance of 26.9 cm from its center. Dizzy's mass is 71.6 g, and the coefficient of static friction between his feet and the turntable is 0.42. The acceleration of gravity is 9.8 m/s2 . Find the maximum number of revolutions per second the turntable can have if Dizzy is to remain stationary relative to the turntable. Correct answer: 0.622561 rev/s (tolerance 1 %). Explanation: The force of static friction is what keeps Dizzy from flying off the turntable by providing the centripetal force which keeps him going in a circular path of radius 26.9 cm . m v2 centripetal The force should not exceed r the maximum force of static friction. This gives
max 2 Ff = s m g = m r max
Question 18 part 1 of 2 10 points A flywheel with a very low friction bearing takes 1.4 hours to stop after the motor power is turned off. The flywheel was originally rotating at 32 rpm. What was the initial rotation rate in radians per second? Correct answer: 3.35103 rad/s (tolerance 1 %). Explanation: = 2f = 3.35103 rad/s. Question 19 part 2 of 2 10 points How many revolutions does the flywheel make before it stops? Correct answer: 1344 revs (tolerance 1 %).
Hence, the maximum speed Dizzy can have without flying off the turntable follows from
2 max =
max
s g r s g = r (0.42) (9.8 m/s2 ) (0.269 m) 1 rev 2 rad
=
Explanation: Let us first find the angular acceleration of the flywheel. It is = -0 = -0.000664887 rad/s2 . t
= (3.91166 rad/s) = 0.622561 rev/s .
The angle rotated is then 1 = 0 t + t2 2 = 8444.6 rad = 1344 revs.
Question 21 part 2 of 2 10 points The turntable starts from rest at t = 0, and has a uniform acceleration of 1.17 rad/s2 . Find the time at which Dizzy begins to slip.
homework 09 FRENNEA, KYLE Due: Mar 28 2007, 4:00 am Correct answer: 3.3433 s (tolerance 1 %). max - 0 (3.91166 rad/s) - 0 = (1.17 rad/s2 ) = 3.3433 s .
8
Explanation: t=
Explanation: Basic Concepts: The moment of inertia, the center of mass and the energy conservation. The rotational kinetic energy is Krot = 1 I 2 . 2 Solution: The moment of inertia of the rod, Irod , with respect to the pivot point is Irod = 1 m L2 , 3
Question 22 part 1 of 4 10 points Consider a rod of length L and mass m which is pivoted at one end. An object with mass m is attached to the free end of the rod. The acceleration of gravity g = 9.8 m/s2 . Note: Contray to the diagram shown below, consider the mass at the end of the rod to be a point particle.
and the moment of inertia Im of the mass m with respect to the pivot point is Imass = m L2 . Then, the moment of inertia of the system I is I = Irod + Im 1 = m L2 + m L2 3 4 = m L2 . 3 Question 23 part 2 of 4 10 points Note: The length C in the figure represents the location of the center-of-mass of the rod plus mass system (but is not drawn to scale). Determine the position of the center of mass from the pivot point; i.e., find C . 1. C = L 2. C = 5 L 8 7 3. C = L 8 4. None of these. 3 L correct 4 1 6. C = L 2 5. C = Explanation: 1 The center of mass of the rod is L. Then, 2 the center of mass of the rod plus mass system
m C L
23 m
Determine the moment of inertia, I, of the system with respect to the pivot point. 1. I = 4 m L2 correct 3 13 2. I = m L2 12
3. I = L2 4. I = 5 m L2 3 5 5. I = m L2 4 3 6. I = m L2 2 7. None of these.
homework 09 FRENNEA, KYLE Due: Mar 28 2007, 4:00 am is 1 L+L m 2 C= m+m 1 1 = L+ L 4 2 3 = L. 4 Question 24 part 3 of 4 10 points The unit is released from rest in the horizontal position. What is the kinetic energy of the unit when the rod momentarily has a vertical orientation? 1. K = 1 mgL 2 Question 25 part 4 of 4 10 points
9
Given: The kinetic energy is KV at the vertical position and the moment of inertia of the system is I. Find the angular velocity of the system in terms of KV and I. 1. = 2 KV correct I
2. None of these. 3. = KV I 2 KV 4. = I KV 2I
5. = 6. =
2. K = 2 m g L 3. K = m g L 5 mgL 2 3 5. K = m g L correct 2 4. K = 6. None of these. Explanation: The potential energy, U , released can be computed in two ways. Method 1: U = U = mg
cm of rod
KV I Explanation: The rotational kinetic energy of the system is given by 1 KV = I 2 , 2 where is the angular velocity of the system. Then, 2 KV = . I Question 26 part 1 of 2 10 points A uniform flat plate of metal is situated in the reference frame shown in the figure below. 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 x (m) 9 10 (m)
+ U
mass
3 = mgL. 2 Method 2:
L + mgL 2
U |rod+m = U |cm of = (2 m) g = 3 mgL. 2
rod+m
3 L 4
y
Calculate the x coordinate of the center of
homework 09 FRENNEA, KYLE Due: Mar 28 2007, 4:00 am mass of the metal plate. Correct answer: 1 m (tolerance 1 %). Explanation: Basic Concept: The center of mass xcoordinate is x dm x where m m ,
x2
10
=
0
x (x - x2 ) dx
x2 0
(x - x2 ) dx
x2
dm , and dm = y dx , where mass of the plate. is the areal density area Solution: Let (x1 , y1 ) = (0 m, 0 m) (x2 , y2 ) = (3 m, 0 m) (x3 , y3 ) = (0 m, 4 m) . The equation for the hypotenuse is y3 - y2 y - y2 = . x - x2 x3 - x 2 The slope of the hypotenuse is s= y3 - y2 x3 - x2 4m-0m 4 = =- . 0m-3m 3
1 3 1 x - (x2 ) x2 2 = 3 1 2 x - (x2 ) x 2 1 3 1 x2 - (x2 ) x2 2 2 = 3 1 2 x - (x2 ) x2 2 2 x3 = 22 3 x2 1 = x2 3 1 = (3 m) 3 = 1 m.
0
(3)
Vertical Center-of-Mass: Using Eq. 3, the y-coordinate of the center of mass of the metal plate is y= 1 y3 3 1 = (4 m) 3 = 1.33333 m .
(1)
Rewriting the equation, we have y = s (x - x2 ) + y2 4 (x - 3 m) + 0 m = - 3 = s (x - x2 ) , Question 27 part 2 of 2 10 points Given: The mass of the plate is m = 1 kg . Calculate the moment of inertia of the triangle with the y-axis as the axis of rotation. Correct answer: 1.5 kg m2 (tolerance 1 %). Explanation: m Areal Density: The areal density = A of the plate is = m
x2
(2)
where y2 = 0 . The x-coordinate of the center of mass is
x2
xcm =
x1 x2 x1
x y dx
x2
y dx x s (x - x2 ) dx
x2
y dx
x1
=
x1
= s
0
m
x2
x1
s (x - x2 ) dx
(x - x2 ) dx
homework 09 FRENNEA, KYLE Due: Mar 28 2007, 4:00 am
2 1 2 s x - (x2 ) x 2 0 m = y3 1 2 - x - (x2 ) x2 x2 2 2 m m = = 1 y3 2 1 x x2 y 3 2 x2 2 2 2m (4) = x2 y 3 2 (1 kg) = (3 m) (4 m) = 0.166667 kg/m2 , y3 from Eq. 1. The area of a where s = - x2 1 1 triangle is A = (base) (height) = x2 y3 , 2 2 which agrees with Eq. 4. Conventional Solution: The moment of x=x inertia Iy cm of the triangle about a y-axis through its center of mass x-coordinate can be accomplished in two steps.
11
=
m
x
= s
1 4 1 x2 - [2 xcm + x2 ] x3 2 4 3 1 + x2 + 2 xcm x2 x2 2 2 cm -x2 x2 cm 2 -
y3 1 4 - x x2 12 2 1 1 + xcm x3 - x2 x2 2 3 2 cm 2 1 2 2 =m x - xcm x2 + x2 cm 6 2 3 1 2 2 2 1 2 x - x + x =m 6 2 9 2 9 2 1 m x2 (5) = 2 18 1 = (1 kg) (3 m)2 18 = 0.5 kg m2 . = using Eq. 1 for s , Eq. 4 for , and Eq. 3 for xcm . Using the parallel axis theorem, we have
x=0 x=x Iy = Iy cm + m x2 cm 1 m x2 + m x2 = 2 cm 18 1 2 1 2 =m x + x 18 2 9 2 1 = m x2 2 6 1 = (1 kg) (3 m)2 6
2m x2 y 3
x=x First: The moment of inertia Iy cm about its center of mass (xcm , ycm ) is determined.
Second: The parallel axis theorem is used. The moment of inertia about the center of x=x mass Iy cm is
x=x Iy cm = x2 x1
(x - xcm )2 dm
x2 x1
= = s
0
(x - xcm )2 y dx
x2
(x - xcm ) (x - x2 ) dx
x2
2
= 1.5 kg m2 . Standard Solution: The moment of inerx=0 tia Iy of the triangle about the x-coordinate is
x=0 Iy x2
= s
0
x3 - [2 xcm + x2 ] x2 +[x2 + 2 xcm x2 ] x cm -[x2 x2 ] cm dx
x2 dm
x1 x2
= s
1 4 1 x - [2 xcm + x2 ] x3 4 3 1 + [x2 + 2 xcm x2 ] x 2 cm
x2 0
=
x1
x2 y dx
x2
= s
0
x2 (x - x2 ) dx 1 4 1 x - (x2 ) x3 4 3
x2 0
-[x2 x2 ] x cm
= s
homework 09 FRENNEA, KYLE Due: Mar 28 2007, 4:00 am 1 4 1 4 x - x 4 2 3 2 y3 1 2m - - x4 = x2 y 3 x2 12 2 1 = m x2 2 6 1 = (1 kg) (3 m)2 6 = s = 1.5 kg m2 , using Eq. 1 for s and Eq. 4 for . Note: This problem has a different triangle for most students. Question 28 part 1 of 2 10 points The figure below shows a rigid 3-mass system which can rotate about an axis perpendicular to the system. The mass of each connecting rod is negligible. Treat the masses as particles. The x-axis is along the horizontal direction with the origin at the left-most mass 2 kg. 2 kg 5 kg 3 kg (3 kg)(2 (3 m) (2 kg) + (5 kg) + (3 kg) 11 kg = (3 m) 10 kg = 3.3 m . + Question 29 part 2 of 2 10 points
12
Note: The position x0 about which the rigid body rotates is not necessarily to scale. x0 2 kg 5 kg 3 kg
L
x
L
3m
x
3m
The masses are separated by rods of length 3 m, so that the entire length is 2 (3 m). Determine the x-coordinate of the center of mass for the three-mass system with respect to the origin. Correct answer: 3.3 m (tolerance 1 %). Explanation: First find the center of mass. Define the origin to coincide with the far left mass M . XCM mi xi = mi (2 kg)(0 ) = (2 kg) + (5 kg) + (3 kg) (5 kg)(1 (3 m) + (2 kg) + (5 kg) + (3 kg)
Explanation: The moment of inertia of a system of point 2 particles is given by I = mi ri . Label the three moments of inertia as Ileft , Imiddle , and Iright . Remembering that the distances ri are with respect to the axis of rotation, we have Ileft Imiddle 1 = (2 kg) (3 m) 3 2 = (5 kg) (3 m) 3 5 (3 m) 3
2
Find the moment of inertia of the 3-mass system about a rotation axis perpendicular to the x-axis and passing through the point (3 m) x0 = , with respect to the origin at the 3 left-most mass 2 kg. Correct answer: 97 kg m2 (tolerance 1 %).
2
2
Iright = (3 kg) Hence
.
I = Ileft + Imiddle + Iright 1 (3 m)2 = (2 kg) 9 4 (3 m)2 + (5 kg) 9
homework 09 FRENNEA, KYLE Due: Mar 28 2007, 4:00 am + (3 kg) = 25 9 (3 m)2
13
97 kg (3 m)2 9 = 97 kg m2 . Question 30 part 1 of 1 10 points
negligible mass) a distance 13 m from the center of mass of the rod. Initially the rod makes an angle of 44 with the horizontal. The rod is released from rest at an angle of 44 with the horizontal, as shown in the figure below The acceleration of gravity is 9.8 m/s2 . Hint: The moment of inertia of the rod 1 about its center-of-mass is Icm = m 2 . 12
A spherical object (with non-uniform density) of mass 14 kg and radius 0.26 m rolls along a horizontal surface at a constant linear speed without slipping. The moment of inertia of the object about a diameter is Icm = (0.33) M R2 . The object's rotational kinetic energy Krot about its own center is what fraction Ktotal of the object's total kinetic energy? Correct answer: 0.24812 (tolerance 1 %). Explanation: We begin by using
44
1.1 kg O 13 m What is the angular speed of the rod at the instant the rod is in a horizontal position? Correct answer: 0.983243 rad/s (tolerance 1 %). Explanation: Let : = 13 m , d = = 13 m , = 44 , and m = 1.1 kg . Basic Concepts: 1 I 2 2 =rF =I KR = 13 m
v = R, where R is the radius of the sphere. Then, Krot = 1 Icm 2 2 1 = (0.33) M R2 2 2 = (0.165) M v 2
The total kinetic energy is given by Ktotal = Krot + Kcm 1 1 = Icm M v 2 + M v 2 2 2 2 = (0.165) M v + (0.5) M v 2 = (0.665) M v 2 Krot (0.165) M v 2 = Ktotal (0.665) M v 2 = 0.24812 . Question 31 part 1 of 1 10 points A uniform rod of mass 1.1 kg is 13 m long. The rod is pivoted about a horizontal, frictionless pin at the end of a thin extension (of
Ki + Ui = Kf + Uf . Solution: I = Icm + m d2 1 = m 2 + m 2 12 13 m 2 = 12 13 (1.1 kg) (13 m)2 = 12 = 201.392 kg m2 .
(1)
homework 09 FRENNEA, KYLE Due: Mar 28 2007, 4:00 am Since the rod is uniform, its center of mass is located a distance from the pivot. The vertical height of the center of mass above horizontal is sin . Using conservation of energy and substituting I from Eq. 1, we have Kf = Ui 1 I 2 = m g sin 2
14
13 m 2 2 = m g sin 24 24 g sin 2 = 13 24 g sin = 13 = 24 (9.8 m/s2 ) sin(44 ) 13 (13 m)
(2)
= 0.983243 rad/s .

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oldmidterm 04 FRENNEA, KYLE Due: Apr 28 2007, 4:00 am Question 1 part 1 of 1 10 points Given: G = 6.6726 10-11 N m2 /kg2 Three 3 kg masses are located at points in the x-y plane as shown in the figure. and has magnitude mm f2 = G 2 y G m2 = y2 (6.

University of Texas - PHY - 303K

homework 03 FRENNEA, KYLE Due: Feb 7 2007, 4:00 am Question 1 part 1 of 2 10 points A descent vehicle landing on the moon has a vertical velocity toward the surface of the moon of 28 m/s. At the same time, it has a horizontal velocity of 46 m/s. At

University of Texas - PHY - 303K

quiz 03 VENNES, ROSS Due: Nov 14 2007, 10:00 pm Question 1, chap 9, sect 1. part 1 of 1 10 points Given: G = 6.673 10-11 N m2 /kg2 Two balls, each with a mass of 0.885 kg, exert a gravitational force of 8.30 10-11 N on each other. How far apart a

University of Texas - PHY - 303K

homework 01 FRENNEA, KYLE Due: Jan 25 2007, 4:00 am1

University of Texas - PHY - 303K

homework 13 VENNES, ROSS Due: Nov 30 2007, 1:00 am Question 1, chap 17, sect 2. part 1 of 1 10 points A rifle is fired in a valley with parallel vertical walls. The echo from one wall is heard 1.04 s after the rifle was fired. The echo from the oth

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homework 09 VENNES, ROSS Due: Nov 2 2007, 1:00 am Question 1, chap 12, sect 2. part 1 of 2 10 points A record has an angular speed of 44.2 rev/min. What is its angular speed? Correct answer: 4.62862 rad/s (tolerance 1 %). Explanation: 1 rev = 6.28

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quiz 01 VENNES, ROSS Due: Sep 19 2007, 10:00 pm Question 1, chap 1, sect 6. part 1 of 1 10 points A plastic tube allows a flow of 13.6 cm3 /s of water through it. How long will it take to fill a 204 cm3 bottle with water? 1. 13.6538 s 2. 14.0881 s

University of Texas - PHY - 303K

homework 11 VENNES, ROSS Due: Nov 16 2007, 1:00 am Question 1, chap 15, sect 2. part 1 of 1 10 points The equation of motion of a simple harmonic oscillator is d2 x = -9 x , dt2 where x is displacement and t is time. What is the period of oscillati

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homework 10 VENNES, ROSS Due: Nov 9 2007, 1:00 am Question 1, chap 11, sect 3. part 1 of 1 10 points A uniform disk of radius 4.7 m and mass 9.1 kg is suspended from a pivot 0.799 m above its center of mass. The acceleration of gravity is 9.8 m/s2

University of Texas - PHY - 303K

quiz 02 VENNES, ROSS Due: Oct 17 2007, 10:00 pm Question 1, chap 6, sect 1. part 1 of 1 10 points A box weighing 660 N is pushed along a horizontal floor at constant velocity with a force of 240 N parallel to the floor. What is the coefficient of k

University of Texas - PHY - 303K

homework 02 FRENNEA, KYLE Due: Jan 31 2007, 4:00 am 2 m/s2 ), Question 1 part 1 of 4 10 points Consider the plot below describing the acceleration of a particle along a straight line with an initial position of -32 m and an initial velocity of -5 m

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homework 12 VENNES, ROSS Due: Nov 21 2007, 1:00 am Question 1, chap 16, sect 2. part 1 of 1 10 points What is the frequency corresponding to a period of 0.01 s? Correct answer: 100 Hz (tolerance 1 %). Explanation: Let : t = 0.01 s . 1 1 = 100 Hz .

University of Texas - PHY - 303K

homework 08 VENNES, ROSS Due: Oct 26 2007, 1:00 am Question 1, chap 11, sect 4. part 1 of 1 10 points An open cart on a level surface rolls without frictional loss through a downpour of rain. The rain falls vertically downward as shown below. As th

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homework 07 VENNES, ROSS Due: Oct 19 2007, 1:00 am %). Question 1, chap 9, sect 1. part 1 of 1 10 points According to some nineteenth-century geological theories (now largely discredited), the Earth has been shrinking as it gradually cools. If so,

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homework 01 VENNES, ROSS Due: Sep 7 2007, 1:00 am Question 1, chap 1, sect 6. part 1 of 1 10 points A piece of pipe has an outer radius, an inner radius, and length as shown in the figure below. Question 2, chap 1, sect 6. part 1 of 2 10 points1

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homework 06 VENNES, ROSS Due: Oct 12 2007, 1:00 am Question 1, chap 8, sect 1. part 1 of 1 10 points The figure below shows a rough semicircular track whose ends are at a vertical height h. A block placed at point P at one end of the track is relea

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homework 04 VENNES, ROSS Due: Sep 28 2007, 1:00 am Question 1, chap 5, sect 1. part 1 of 2 10 points Four forces act on a hot-air balloon, as shown from the side. 264 N 264 N39 2 N1118 N252 N 118 N 115 N 252 N 115 NScale: 100 N Basic Conce

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homework 03 VENNES, ROSS Due: Sep 21 2007, 1:00 am Question 1, chap 4, sect 3. part 1 of 2 10 points A particle travels horizontally between two parallel walls separated by 18.4 m. It moves toward the opposing wall at a constant rate of 7.6 m/s. Al

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homework 05 VENNES, ROSS Due: Oct 5 2007, 1:00 am Question 1, chap 7, sect 1. part 1 of 2 10 points A 15.5 kg block is dragged over a rough, horizontal surface by a constant force of 177 N acting at an angle of 32.8 above the horizontal. The block

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midterm 02 FRENNEA, KYLE Due: Mar 7 2007, 11:00 pm1Mechanics - Basic Physical ConceptsMath: Circle: 2 r, r2 ; Sphere: 4 r2 , (4/3) r3 b2 Quadratic Eq.: a x2 + b x + c = 0, x = -b 2 a-4 a cCartesian and polar coordinates: y x = r cos , y

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homework 07 FRENNEA, KYLE Due: Mar 7 2007, 4:00 am Question 1 part 1 of 1 10 points On the way from a planet to a moon astronauts reach a point where that moon's gravitational pull is stronger than that of the planet. The masses of the planet and t

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homework 02 VENNES, ROSS Due: Sep 14 2007, 1:00 am For the entire trip, Question 1, chap 2, sect 1. part 1 of 1 10 points The average speed of an orbiting space shuttle is 21900 mi/h. The shuttle is orbiting about 227 mi above the Earth's surface.

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Conclusion: In this lab, we determined the speed of a glider as it slid down a frictionless incline. By observing the initial speed, the final speed, and the time taken to complete this path we can determine the acceleration of the cart, which should

UCLA - CS - 151A

BOOLEAN ALGEBRAS Important class of algebras extensively used for many purposes1 Basis of the switching algebra (SA) for formal treatment of switching circuits: Transformation of switching expressions Identities from BA enable graphical and ta

Kent State - PHYSICS - 11030

Kent State - PHYSICS - 11030

Kent State - PHYSICS - 11030

Kent State - PHYSICS - 11030

Kent State - PHYSICS - 11030

Cincinnati - CHEM - 101

Cincinnati - ECEE - 120

Benjamin Jones 03/09/08 Final Report Laboratory 9 Header File:/ / / / / / / / / / / / / / / / / / * File: mysongs.h Song Library - Keeps track of a library of songs. Reads song data from file and stores each song record in a linked list. User can ch

Cincinnati - ECEE - 120

University of Cincinnati Department of Computer Science CS 121-002 Computer Science 1Laboratory 2 Final ReportBenjamin D. Jones 16 January 20081. IntroductionThe purpose of this lab assignment is to learn how to develop C+ programs that use sim

Cincinnati - ECEE - 120

5.4 1. This is an infinite loop, and the "F" in for should not be capitalized. 2. The switch cases are without "break" statements. 3. This is an infinite loop that will display odd integers greater than 19. 4. The variable "counter" is not defined, t

Cincinnati - CHEM - 101

Cincinnati - MATH - 253

15 Math 253-007: CALCULUS III (Spring 2008)Instructor: Dr. Mihaela Poplicher Office: 811E Old Chem Office hours: MTF: 11-12noon (or by appointment) Phone: 513-556-1223 E-mail: Mihaela.Poplicher@uc.edu Textbook: "Calculus: Concepts and Contexts", Th

Cincinnati - ECEE - 120

Benjamin Jones February 7, 2008 Final Report Laboratory 5The Program/ * / File: bankaccount.cpp / / Compute Savings Interest - User inputs an interest rate as an integer as an / // / / / / Author: Benjamin Jones / Created: 07 Febuary 2008 / Update

Cincinnati - ECEE - 120

4.31 z = x + y; cout < +z; 4.27/ * / File: bintodec.cpp / / Binary to Decimal - User inputs an integer as a binary number, / / / Author: Benjamin Jones / Created: 28 January 2008 / Updated: Original / / Assignment: 4.27 / Compiler: MinGW32 G+ 3.4.2

Cincinnati - CLAS - 356

Holt N. Parker Sex & Gender in the Classical World15 CLAS 356Syllabus This class is an exploration of the sex and gender system of Greece and Rome. It is a detailed look at one aspect As a 300-level class it demandsof ancient civilization.a gr

Cincinnati - ECEE - 120

University of Cincinnati Department of Computer Science CS 121-002 Computer Science 1Laboratory 3 Design ReportBenjamin Jones 22 January 20081. IntroductionThe purpose of this laboratory is to learn how to utilize input/output statements, expre

Cincinnati - CHEM - 101

Cincinnati - ECEE - 120

4.31 z = x + y; cout < +z; 4.27/ * / File: bintodec.cpp / / Binary to Decimal - User inputs an integer as a binary number, / / / Author: Benjamin Jones / Created: 28 January 2008 / Updated: Original / / Assignment: 4.27 / Compiler: MinGW32 G+ 3.4.2

Cincinnati - ECEE - 120

Benjamin Jones 02/14/08 Final Report Laboratory 6 Program Source Code Header File:/ * / File: findroots.h / / Stats of a list of data - User enters a number to take a root of, the root / to take of that number, a desired precision, and the program w

Cincinnati - ECEE - 120

University of Cincinnati Department of Computer Science CS 121-002 Computer Science 1Laboratory 3 Final ReportBenjamin Jones 24 January 20081. IntroductionThe purpose of this laboratory is to learn how to utilize input/output statements, expres

Cincinnati - ECEE - 120

Benjamin Jones 31 January 2008 Final Report Laboratory 4 Program Header file "stats.h":#include <cstdlib> #include <iostream> #include <cmath>using namespace std;const int OK = 0; const double HALF = 0.5; const int NOTOK = 1; const double MAX_DI

Cincinnati - ECEE - 120

Benjamin Jones 02/28/08 Final Report Laboratory 8 Header File:/ * / File: functions.h / / Matrix Multiply - User creates a file that specifies matrix dimensions, / and matrix data, and the program multiplies that matrices together, / outputting the

Cincinnati - ECEE - 120

Benjamin Jones 02/26/08 Design Report Laboratory 8 I. Get necessary input I.A Open the input text file I.B Get dimensions for matrix one and two I.B.1 Read first line of file: set first number to first matrix's rows, set second number to first matrix

Cincinnati - ECEE - 151

EECE 151: Introduction to ECE LaboratoryLaboratory 2:Introduction to Digital Signal Processing and CommunicationsDescription: This week's exercises will provide you with an introductory understanding of signal processing (in particular, digital

Cincinnati - ECEE - 120

Test Number 1Input 234, 54, 468, 38, 342225.32, 21.89, 29.72, 45.1, 0.00000369Expected Output Average: 227.2 Variance: 34251.2 Std Dev: 185.071 Max: 468 Min: 38 Theo Avg: 253 Theo Var: 15408.3 Theo Dev: 124.13 Avg Diff: -25.8 Var Diff: 18842.

Cincinnati - ECEE - 120

Benjamin Jones January 29, 2008 Design Report Laboratory 4 Main Module I. a. Get input from user Begin loop while current input number (floating point) is non-negative 1. Add current number to the total summation of numbers (for average) 2. Add the s

Cincinnati - ECEE - 120

Benjamin Jones 02/12/08 Design Report Laboratory 6Primary Organization 1. Get necessary input from user 2. Calculate the nth root of a value to a given precision 3. Output the nth root of that value when a precision is met Modular Organization 1. G

Cincinnati - ECEE - 120

University of Cincinnati Department of Computer Science CS 121-002 Computer Science 1Laboratory 2 Design ReportBenjamin D. Jones 16 January 20081. IntroductionThe purpose of this lab assignment is to learn how to develop C+ programs that use si

Cincinnati - ECEE - 120

Benjamin Jones 02/19/2008 Design Report Laboratory 7 *Note: This will greate a *true* histogram graph of roots I. Get necessary input from user A. Get number to take roots of (must be 0<x<100) and of type double 1. Read input 2. Validate input, if no

Cincinnati - ECEE - 120

Benjamin Jones 06 February 2008 Design Report Laboratory 5 Get the interest from the user expressed as a percent (a floating point number) a. Validate the input (make sure it is non-negative, and less than 100) 1. If input is greater than 100 or less

Cincinnati - ECEE - 120

Example Output

Cincinnati - PHYS - 201

Is Female to Male as Nature Is to Culture? Sherry B. Ortner Feminist Studies, Vol. 1, No. 2. (Autumn, 1972), pp. 5-31.Stable URL: http:/links.jstor.org/sici?sici=0046-3663%28197223%291%3A2%3C5%3AIFTMAN%3E2.0.CO%3B2-L Feminist Studies is currently pu

Cincinnati - CHEM - 101

Cincinnati - CHEM - 101

Cincinnati - ECEE - 151

Laboratory #1 Crystal Radio Autumn 2007Description: This week's lab will provide you with an understanding of how to build and modify a crystal radio set. In addition to learning about this popular hobby, you will be introduced to one of the oldest

Cincinnati - ECEE - 120

Benjamin Jones 03/04/08 Design Report Laboratory 10High Level Design I. Initialize node structure for linked list II. Populate data into the list from file 1. Ask user which file to load 2. If file does not exist a. Ask user to create file b. Ask u