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Unformatted text preview: The coordinates of the cm are given by:
x cm = i i i and y cm = i i i
M ∴ x cm = (15.0 × 0.5) + (5.00 × 1.5) + (15.0 × 2.5)
= 1.50 m .
i.e., x cm =
35.0 DISCUSSION PROBLEM [8.1]:
Consider a 3m × 3m sheet with a 2 m × 1m region of
negative mass (shaded). By symmetry, the cm of the
whole sheet is at (1.5,1.5),
y 1m 1m and the shaded region ~ Surprise, surprise ?? is at
o also (15.0 × 1.5) + (5.00 × 0.5) + (15.0 × 1.5)
y cm =
= 1.36 m .
i.e., y cm =
∴ rcm = (1.50 m,1.36 m) ,
from the origin chosen.
DISCUSSION PROBLEM [8.1]: 3m • 2m (1.5, 2.0).
3m The mass of the whole 1m sheet is
× 35 = 45.0 kg,
and the mass of the shaded region is × 35 = 10.0 kg.
45.0 × 1.5 + ( −10.0) × 1.5
∴ x cm =
= 1.50 m
45.0 + ( −10.0)
(0,0) x and Can you think of another way with less calculations?
(HINT: not by choosing another origin but what about
negative mass?) y cm = 45.0 × 1.5 + ( −10.0) × 2
= 1.36 m,
45.0 + ( −10.0) as before! 1 DISCUSSION PROBLEM [8.2]: DISCUSSION PROBLEM [8.2]:
cm A canoeist drops a paddle in the lake. Assuming both
the canoe and paddle are stationary, can he retrieve the
paddle by moving towards the bow? ... Why ... or why
not? No, he cannot retrieve the paddle! Since there are no
external forces, the center of mass of the canoe plus
canoe-ist remains fixed ... Newton’s 1st Law! (Of
course, we assume that the canoe is stationary.)
PS: Don’t test this answer unless you have a spare
paddle in the canoe! 2 DISCUSSION PROBLEM [8.3]:
DISCUSSION PROBLEM [8.3]: (a)
(a) (b) (b) No, there is no violation! The system is actually the A cheetah crouches, motionless, behind a bush (a). It cheetah and the Earth ... the Earth is essential for the spots an antelope and accelerates after its prey (b). cheetah to move (by a friction force)! Conservation of Clearly, at (a) the cheetah’s momentum is zero whereas momentum requires that the gain in momentum of the at (b) it has finite momentum. cheetah (to the right) is exactly equal and opposite to the
gain in momentum of the Earth (to the left). Since the I thought momentum was supposed to be conserved ... mass of the Earth ( 6 × 1024 kg) is many orders of does this scenario violate the law of conservation of magnitude greater than that of the cheetah ( ~ 102 kg), momentum? Explain. the velocity of the Earth is correspondingly ( ~ 10 −22)
orders of magnitude smaller than that of the cheetah! 3 ...
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