A 100 kg load is placed on a 40 kg stretcher to be carried by two persons, one on each end. The stretcher is 2 m long and the load is placed 0.66 m from the left end. How much force must each person exert to carry the stretcher?
A stationary space station can be approximated as a hollow spherical shell of
mass 6 tons (6000 kg) and inner and outer radii of 5 m and 6 m. To change its
orientation, a uniform fly wheel of radius 10 cm and mass 10 kg located at the
center of the station is spun quickly from rest to 1000 rpm.
(a) How long (in minutes) will it take the station to rotate by 10o?
(b) What energy (in Joules) is needed for the whole operation?
(Moment of Inertia of hollow spherical shell: [2M(Ro5 - Ri5 )]/[5(Ro3 - Ri3)] where Ro is outer radius and Ri is inner radius)
Two identical billiard balls of radius R and mass M, rolling with CM velocities
±vi, collide elastically, head-on.
Assume that after the collision they have both reversed motion and are
(a) Find the impulse which the surface of the table must exert on each ball during its reversal of motion.
(b) What impulse is exerted by one ball on the other?
FballΔt = -(2mv + 4mv/5)i = -(14mv/5)
i is the impulse the right ball exerts on the left ball.
The impulse the left ball exerts on the right ball is (14mv/5) i.
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
(a) Calculate the principal moments of inertia of the spaceship about its CM 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.
Details of the calculation:
(a) Use cylindrical coordinates, with the origin at the origin of the primed system.
I3' = ∫(x2 + y2) dm = ∫ρ2μdV.
Here ρ is a cylindrical coordinate and μ is the mass per unit volume.
m = μ∫0hdz π(rz/h) = μπr2h/3.
I3' = μ∫02πdφ ∫0rdρ ∫hρ/rhdz ρ3 = 2πμ∫0rdρ ρ3(h - hρ/r)
= 2πμh[r4/4 - r4/5] = πμhr4/10 = 3mr2/10.
I1' = I2' from symmetry.
I2' = ∫(x2 + z2) dm = ∫(ρ2cos2φ + z2)μdV = ∫ρ2cos2φ μdV + ∫z2μdV.
I2' = μ∫02πcos2φdφ ∫0rdρ ∫hρ/rhdz ρ3 + μ2π∫0rdρ ∫hρ/rhdz ρ z2
= πμhr4/20 + πμh3r4/5 = (3/5)m(r2/4 + h2) = I1'.
The CM is located a ρ = 0 and zCM, where
zCM = (μ/m)∫02πdφ ∫0rdρ ∫hρ/rhdz ρ z = 3h/4.
Using the parallel axes theorem we therefore have
I3 = I3' = 3mr2/10,
I1 = I1' - m(3h/4)2 = (3/20)mr2 + (3/80)mh2 = I2.
(b) L1 = I1ω1, L2 = I1ω2,
L3 = I3ω3.
We can chose the orientation of our axes so that L2 = 0, ω2 = 0.
In the body-fixed coordinate system L, ω, and the x3-axis lie in the same plane.
Consider a point P on the x3-axis of the top, displaced by r from the CM.
With respect to space-fixed coordinates, the velocity of point P is v = ω × r.
At every instant the velocity of points on the x3-axis is perpendicular to the plane containing L, ω, and the x3-axis, so the x3-axis must rotate about L.
The direction of L is fixed in space.
Consider a right triangular lamina of areal density ρ, with one edge of length a along the x-axis and another edge of length b along the y-axis, as shown in the diagram.
(a) Find the center of mass (X,Y,Z) in this coordinate system.
(b) Find the components of the inertia tensor in this coordinate system.
(c) For a = b, transform the inertia tensor to principal axes, giving the angle between the principal axes and those shown in the diagram, and find the three moments of inertia in the principal axes system.
(d) Use the parallel axis theorem to find the principal moments of inertia about the center of mass.