1. Problem 12.131 A 250-g collar can slide on a 400 mm
horizontal rod which is free to rotate
about a vertical shaft. The collar is
initially held at A by a cord attached
to the shaft and compresses a spring
of constant 6 N/m, which is
undeformed when the collar is
located 500 mm from the shaft. As
the rod rotates at the rate qo = 16 rad/s, the cord is cut and the
collar moves out along the rod. Neglecting friction and the
mass of the rod, determine for the position B of the collar (a) the
transverse component of the velocity of the collar, (b) the radial
and transverse components of its acceleration, (c) the
acceleration of the collar relative to the rod. .

2. 400 mm
100 mm A B
Problem
Solving Problems on Your Own
The collar is initially held at A by a
cord attached to the shaft and
compresses a spring. As the rod
rotates the cord is cut and the
collar moves out along the rod to B.
1. Kinematics: Examine the velocity and acceleration of the
particle. In polar coordinates:
v = r er + r q eq
a = (r - r q 2 ) er + (r q + 2 r q ) eq
r = r er eq er q . . . . . . .

3. . Problem 12.131 400 mm 100 mm Solving Problems on Your OwnA B
Problem
Solving Problems on Your Own
The collar is initially held at A by a
cord attached to the shaft and
compresses a spring. As the rod
rotates the cord is cut and the
collar moves out along the rod to B.
2. Angular momentum of a particle: Determine the particle
velocity at B using conservation of angular momentum. In polar
coordinates, the angular momentum HO of a particle about O is
given by
HO = m r vq
The rate of change of the angular momentum is equal to the sum
of the moments about O of the forces acting on the particle.
S MO = HO
If the sum of the moments is zero, the angular momentum is
conserved and the velocities at A and B are related by
m ( r vq )A = m ( r vq )B .

4. 400 mm
100 mm A B
Problem
Solving Problems on Your Own
The collar is initially held at A by a
cord attached to the shaft and
compresses a spring. As the rod
rotates the cord is cut and the
collar moves out along the rod to B.
3. Kinetics: Draw a free body diagram showing the applied
forces and an equivalent force diagram showing the vector
ma or its components.

5. . . Problem 12.131 400 mm Solving Problems on Your Own 100 mm A
The collar is initially held at A by a
cord attached to the shaft and
compresses a spring. As the rod
rotates the cord is cut and the
collar moves out along the rod to B.
4. Apply Newton’s second law: The relationship between the
forces acting on the particle, its mass and acceleration is given
by S F = m a . The vectors F and a can be expressed in terms of
either their rectangular components or their radial and transverse
components. Absolute acceleration (measured with respect to
a Newtonian frame of reference) should be used.
With radial and transverse components:
S Fr = m ar = m ( r - r q 2 ) and S Fq = m aq = m ( r q + 2 r q ) . .

6. . . . . . . . . vq = r q ar = r - r q 2 aq = r q + 2 r q 400 mm 100 mmB
Problem Solution
Kinematics. q . vq .
vq = r q
ar = r - r q 2
aq = r q + 2 r q A B ar r . . aq . . . .

7. . . . m rA (vq )A = m rB (vq )B since (vq )A = rA q (vq )B = q
400 mm
100 mm A B
Problem Solution
(a) The transverse component of
the velocity of the collar.
Angular momentum of a particle.
m rA (vq )A = m rB (vq )B
since (vq )A = rA q
(vq )B = q
(vq )B = (16 rad/s)
(vq )B = 0.4 m/s
(rA)2 rB
( 0.1 m )2
0.4 m . q . .
(vq)A
(vq)B A B r
rA = 0.1 m
rB = 0.4 m

8. 400 mm
100 mm A B
Problem Solution
(b) The radial and transverse
components of acceleration.
Apply Newton’s second law.
Only radial force F (exerted by the
spring) is applied to the collar.
For r = 0.4 m:
F = k x = (6 N/m)(0.5 m m)
F = 0.6 N
Kinetics; draw a free
body diagram. F
m ar
m aq =
SFr = mar: N = (0.25 kg) ar
ar = 2.4 m/s2
+ SFq = maq: = (0.25 kg) aq
aq = 0

9. . . . . . . . . . vq = r q, q = vq q = = 1 rad/s r 0.4 m/s 0.4 m
400 mm
100 mm A B
Problem Solution
(c) The acceleration of the
collar relative to the rod.
Kinematics.
For r = 0.4 m:
vq = r q, q =
q = = 1 rad/s . . vq r .
0.4 m/s q .
0.4 m vq
The acceleration of the collar
relative to the rod is r. A . B ar r . .
ar = r - r q 2
(2.4 m/s2) = r - (0.4m)(1 rad/s)2
r = 2.8 m/s2 . aq .