Most problems have moved to:
| http://electron9.phys.utk.edu/phys513/Modules/module2/problems2.htm |
Additional Problems:
Problem:
A uniform rectangular door of mass m with sides a and b
(b > a) and negligible thickness rotates with constant angular velocity
W about a diagonal. Ignore gravity. Show that the torque
|t| = m(b2 - a2)abW2/(12(b2
+ a2)) must be applied to keep the axis of
rotation fixed.
Solution:
| Concepts, principles, relations
that apply to the problem: Rigid body motion, Euler's equations |
| Why do they apply? Euler's equation for the motion of a rigid body in a force field are
They give us the relationships between torque, angular acceleration and angular velocity about the principal axes. |
|
How do they apply?
W1 =
Wx = Wcosq.
W2 = Wy
= -Wsinq.
W3 = Wz
= 0. dW1/dt = dW2/dt
= dW3/dt =0. |
| Details of the calculation: None |
Problem:
A uniform thin wheel of mass m is attached to a
massless axle, so that the axle is along the symmetry axis. The system now is caused to
rotate about the end point of the axle, which is fixed in space. This axle is described by
the Eulerian angles q and f with
respect to the direction of the total angular momentum L, and the spinning
of the rigid body is described by y. Now consider the physics
of the situation in a coordinate system P attached to the axle with the same
q and f , but with
y = 0. (See figure.) Notice that the system is now neither an inertial system nor a rigid body
system.
(a) What is Euler's equation of motion expressed in terms of the coordinate
system P?
(b) Show that the total angular momentum L is conserved.
(c) Show that the symmetry axis and
w
precess about L in the same plane.
(d) Show that tanq
= (1/2)tana ,
where a = angle between the symmetry axis and
w .
(e) Show that df/dt = (w sina)/sinq .
Solution:
| Concepts, principles, relations
that apply to the problem: Euler's equations, dL/dt|rotating + W ´ L = t, t = 0 --> L = constant |
| Why do they apply? We assume that the system is located in free space, there is no gravitational force acting on the system and the torque is zero. L is constant and the (X,Y,Z) system is fixed in space. The (x.y.z) system can have angular velocity W in the (X,Y,Z) system, and the disk has angular velocity yk in the (x.y.z) system. The rotating coordinate system P (i.e. the (x.y.z) system) is not a body fixed system, but its axes are principal axes. We therefore have Lx = wx, Ly = wy, Lz = wz, where the components are with respect to the P system. We have Ix = Iy = (1/2)Iz. |
| How do they apply? w = W + y. We are adding angular velocity vectors. From geometry we have: wx = Wx = dq/dt, wy = Wy = sinq df/dt, wz = Wz + y = sinq df/dt + y. Lx = Ixdq/dt, Ly = Iysinq df/dt, Lz = Iz(sinq df/dt + y). |
| Details of the calculation: (a) Euler's equations are: Ixdwx/dt + (Iz - Ix)WyWz = 0, Ixdwy/dt + (Ix - Iz)WzWx = 0, Izdwz/dt = 0. Ixdwx/dt + IxWyWz = 0, Ixdwy/dt - IxWzWx = 0, dwz/dt = 0. Remember: Ix = Iy = (1/2)Iz. (b) t = 0 --> L = constant. (c) Ly = Lsinq, Lz = Lcosq, Lx = 0. wy = Lsinq/Ix, wz = Lcosq/(2Ix), wx = 0. In the P system the components of L and w are constant. L and w are constant vectors in the x-y plane. This plane rotates with constant angular velocity df/dt about the Z-axis. Ly = Lsinq = Ix sinq df/dt. df/dt = L/Ix. (d) tana = wy/wz = (Lsinq/Ix)/(Lcosq/(2Ix)) = 2 tanq. (e) wy = sinq df/dt, df/dt = wy/sinq = wsina/sinq. |
Problem:
Suppose a uniform wheel of radius R, thickness d, and
mass M is rotating with uniform angular speed w about an
axis that passes through its center of mass but makes an angle
q
with a line perpendicular to the wheel. Find the angular momentum of the wheel and the
torque on the axis.
Solution:
| Concepts, principles, relations
that apply to the problem: Rigid body motion, Euler's equations |
| Why do they apply? Euler's equations give us the relationships between torque, angular acceleration and angular velocity about the principal axes. |
| How do they apply? Orient the body-fixed coordinate system so that the z-axis is perpendicular to the wheel, and w lies in the yz-plane. Then wx = wsinq, wy = 0, wz = wcosq. Euler's equations are Ixdwx/dt + (Iz - Iy)wywz = tx, Iydwy/dt + (Ix - Iz)wzwx = ty, Izdwz/dt + (Iy - Ix)wxwy = tz. dwx/dt = dwy/dt = dwz/dt = 0. wy = 0. (Ix - Iz)wzwx = ty, (Ix - Iz)w2sinqcosq = ty, [(Ix - Iz)w2(sin2q)/2]j = t = torque on the axis. |
| Details of the calculation: L = iIxwsinq + kIzwcosq. Here i and k refer to the body fixed axis. For a cylinder of radius R, height d, and mass M we have: Iz = M/(pr2d)ò-d/2d/2dzò0R2pr3dr = MR2/2, Ix = M/(pr2d)ò-d/2d/2dzò02pdqò0Rrdr(r2sin2q + z2) = Md2/12 + MR2/4 = Iy. |
Problem:
A rigid body of arbitrary shape has moments of inertia I1,
I2, and I3 about its three principal (body-fixed)
axes. It is set into motion such that its angular momentum L and its kinetic energy
T satisfy the relation L2 = 2I1T.
No
external forces are present. Let w1,
w2, and w3
be the
components of the angular velocity about the principal axes.
(a) What is the general relationship between L2, the moment of
inertia tensor, and the angular velocity? What is the corresponding expression for T?
(b) Write the differential equations describing the behavior of the rigid body.
Divide one equation by another to eliminate one of the components of
w.
Integrate the resulting equations to obtain relationships among the various components of
w.
(c) Use the results of part (b) to integrate the differential equations of
motion. In particular obtain a closed form expression for
w1(t).
(d) Let w1(0) = 0 and sketch a graph of
w1(t) and w2(t).
| Solution
|
Problem:
A rigid, symmetrical spaceship is shaped in the form of a
cone with a uniform density. The height of the cone is h, the radius of the base is
r, and the total mass is m1. Being suspended in outer space
without any external forces acting on it, the space ship has a center of mass velocity v
and angular momentum L not quite parallel to the symmetry axis. Thus it
experiences precession.
(a) Calculate the principal moments of inertia of the spaceship in terms of h,
r, and m1.
(b) Show that the symmetry axis rotates in space about the fixed direction of the
angular momentum L.
(c) The spaceship makes a soft landing onto a stationary space station consisting of
a very thin, uniform disk of radius R and mass m2. The tip of the
conic section touches the outside edge of the disk. After this totally inelastic
collision, describe the motion of the compound object.
Solution:
| Concepts, principles, relations
that apply to the problem: The parallel axes theorem, conservation of energy and angular momentum |
| Why do they apply? We want to find the moments of inertia of a symmetric top about the principal axes through the center of mass. Using the parallel axes theorem makes the calculation of I1 and I2 easier. We first find the moments of inertia about the primed axes and then use the principal axes theorem to find the moments of inertia about the unprimed axes. |
| How do they apply? |
| Details of the calculation: |
|
Student solution:
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