Phys 2101 Gabriela González · 10 Moment of inertia with respect to an arbitrary rotation axis: I...
Transcript of Phys 2101 Gabriela González · 10 Moment of inertia with respect to an arbitrary rotation axis: I...
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Pure rotation of a reference line in a solid body around an axis is described with an angle as a function of time.
θ=s/r ccw: positive cw: negative
radians
rad/sec
rad/sec2
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• Trajectory is a circle (“circular” motion) • Speed is constant (“uniform motion”)… • … but velocity is not constant!?
)(
ˆ)sin(ˆ)cos()(
ˆ)cos()sin()(
ˆ)sin(ˆ)cos()(
2
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txjtRitRta
jtRitRtvjtRitRtx
ω
ωωωω
ωωωω
ωω
−=
−−=
+−=
+=
RRvaRvRs 22
;; ωωθ ====Derive the following formulae:
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rs θ=
rvrdtd
dtds
ωθ
=⇒=
rardtd
dtdv
t αω
=⇒=
If speed is not constant, there is angular and tangential acceleration, apart from radial acceleration.
If speed is constant, there is no angular acceleration, but there still is radial acceleration.
Assume a constant radius: circular motion
rar2ω=
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A wheel A of radius rA = 10 cm is coupled by belt B to wheel C of radius rC = 25 cm. The angular speed of wheel A is increased from rest at a constant rate of 1.6 rad/s2. Find the time for wheel C to reach a rotational speed of 100 rev/min, assuming the belt does not slip. (Hint: If the belt does not slip, the linear speeds at the rims of the two wheels must be equal.)
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Kinetic energy:
( )2
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22442
12332
1222
2112
1
22442
122332
122222
122112
1
2442
12332
12222
12112
1
ω
ω
ωωωω
I
rmrmrmrm
rmrmrmrm
vmvmvmvmK
=
+++=
+++=
+++=
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Moment of inertia with respect to an arbitrary rotation axis: 2MhII com +=
Four identical particles of mass 0.50 kg each are placed at the vertices of a 2.0 m 2.0 m square and held there by four massless rods, which form the sides of the square. What is the rotational inertia of this rigid body about an axis that (a) passes through the center of mass, the midpoints of opposite sides and lies in the plane of the square, (b) passes through two masses on one side?
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The figure shows an overhead view of a meter stick that can pivot about the point indicated, which is to the left of the stick's midpoint. Two horizontal forces, F1 and F2, are applied to the stick. Only F1 is shown. Force 2 is perpendicular to the stick and is applied at the right end. If the stick is not to turn, (a) what should be the direction of F2, and (b) should F2 be greater than, less than, or equal to F1?
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How effective is a force to produce rotation?
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2 3 4
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Torque = Force x distance x sin(angle)
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τ = r × F = rF sinφ
Positive if the force produces a ccw rotation; negative if the force produces a cw rotation.
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The body in Fig. 11-38 is pivoted at O. Three forces act on it in the directions shown: FA = 10 N at point A, 8.0 m from O; FB = 16 N at point B, 4.0 m from O; and FC = 19 N at point C, 3.0 m from O. What is the net torque about O?
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The figure shows the massive shield door at a neutron test facility at Lawrence Livermore Laboratory; this is the world's heaviest hinged door. The door has a mass of 44,000 kg, a rotational inertia about a vertical axis through its huge hinges of 8.7x104 kg·m2, and a (front) face width of 2.4 m. Neglecting friction, what steady force, applied at its outer edge and perpendicular to the plane of the door, can move it from rest through an angle of 90° in 30 s?
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In the figure, a block has mass M = 500 g, the other has mass m = 460 g, and the pulley, which is mounted in horizontal frictionless bearings, has a radius of 5.00 cm. When released from rest, the heavier block falls 75.0 cm in 5.00 s (without the cord slipping on the pulley).
(a) What is the magnitude of the blocks' acceleration?
What is the tension in the part of the cord that supports (b) the heavier block and (c) the lighter block?
(d) What is the magnitude of the pulley's angular acceleration? (e) What is its rotational inertia?
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(a) What is the magnitude of the blocks' acceleration?
What is the tension in the part of the cord that supports (b) the heavier block and
(c) the lighter block?
(Tensions are not equal any more!!!) (d) What is the magnitude of the pulley's angular acceleration?
(e) What is its rotational inertia?
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y =12at 2 ⇒ a = 2y / t 2 = 2 × 0.75m /25s2 = 0.06m /s2
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Mg−T1 = Ma⇒ T1 = M(a + g) = 0.5kg × 9.86m /s2 = 4.93N
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T2 −mg = ma⇒ T2 = m(a − g) = 0.496kg × 9.794m /s2 = 4.86N
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α = a /R = 0.06m /s2 /0.05m =1.2rad /s2
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τ = −T1R + T2R = IαI = (T2 −T1)R /α = 0.07N × 0.05m /1.2rad /s2 = 0.0029kgm2
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A meter stick is held vertically with one end on the floor and is then allowed to fall. Assuming that the end on the floor does not slip… (a) What forces are acting on the stick? (b) What forces are doing work? (c) Find the speed of the other end when it hits the floor.
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A meter stick is held vertically with one end on the floor and is then allowed to fall. Assuming that the end on the floor does not slip… (a) What forces are acting on the stick? Gravity (at the center of mass), static friction and normal force (at bottom end) (b) What forces are doing work? Only gravity does (positive) work.
(c) Find the speed of the other end when it hits the floor. The motion is pure rotation about the lower end. Use conservation of energy:
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Ibot = Icom + M(L /2)2 =1
12ML2 +
14ML2 =
13ML2
12Ibotω
2 −Mg L2
= 0⇒ω =MgLIbot
= 3 gL
vtop =ωL = 3gL = 5.4 m/s