Phys 2101 Gabriela González · 10 Moment of inertia with respect to an arbitrary rotation axis: I Mh2 com + Four identical particles of mass 0.50 kg each are placed at the vertices

Phys 2101 Gabriela González

2

3

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

4

221

00

0

attvxxatvv

++=

+=2

21

00

0

ttt

αωθθ

αωω

++=

+=

x θ

x

y

5

•  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

22

txjtRitRta

jtRitRtvjtRitRtx

ω

ωωωω

ωωωω

ωω

−=

−−=

+−=

+=

RRvaRvRs 22

;; ωωθ ====Derive the following formulae:

6

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=

7

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.)

8

Kinetic energy:

( )2

21

22442

12332

1222

2112

1

22442

122332

122222

122112

1

2442

12332

12222

12112

1

ω

ω

ωωωω

I

rmrmrmrm

rmrmrmrm

vmvmvmvmK

=

+++=

+++=

+++=

9

10

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?

11

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?

12

How effective is a force to produce rotation?

1

2 3 4

5

6

Torque = Force x distance x sin(angle)

€

τ = r × F = rF sinφ

Positive if the force produces a ccw rotation; negative if the force produces a cw rotation.

13

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?

14

⇒= amF

tt

rr

maFmaF

=

=

rmFrmTF

αφ

ωφ

=

=−

sincos 2

ατ

αφτ2

2sinrm

rmFr=

==

ατ I=

15

We know all of that already!!

16

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?

17

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?

18

(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?

€

y =12at 2 ⇒ a = 2y / t 2 = 2 × 0.75m /25s2 = 0.06m /s2

€

Mg−T1 = Ma⇒ T1 = M(a + g) = 0.5kg × 9.86m /s2 = 4.93N

€

T2 −mg = ma⇒ T2 = m(a − g) = 0.496kg × 9.794m /s2 = 4.86N

€

α = a /R = 0.06m /s2 /0.05m =1.2rad /s2

€

τ = −T1R + T2R = IαI = (T2 −T1)R /α = 0.07N × 0.05m /1.2rad /s2 = 0.0029kgm2

19

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.

20

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:

€

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