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PHYS 230
Fall 2007
Assignment 2
This assignment is about accelerated (noninertial) reference frames. Unless otherwise
speciﬁed, solve these problems in the appropriate accelerated frame.
1.
[5 points.]
A boy is riding on a ﬂat railcar, on level ground, that has an acceleration
a
in the direction of its motion. (It is speeding up!) At what angle to the vertical
should he toss a ball so that he can catch it without shifting his position on the car?
Is the angle forwards or backwards?
In the accelerated frame of the railcar, the forces on the ball will be
mg
, downwards,
and

ma
, in the direction opposite to the acceleration
a
. This is as if there is an
eﬀective gravitational force, whose value is
m
√
g
2
+
a
2
, and whose angle to the vertical
is
arctan
a/g
. The ball will therefore “return” directly to the boy if he throws it ex
actly opposite to the direction of this force. (Think of throwing a ball upwards against
“normal” gravity!)
2.
[5 points.]
The “conventional” way to solve this problem takes much longer.
Suppose that the intial velocity of the railcar is
u
. Suppose that the ball has initial
vertical velocity
v
y
and horizontal velocity (which does not change)
v
x
+
u
. Then, the
time to return to the ground (level of the railcar) is
t
= 2
v
y
/g
. In this time, the
horizontal distance traveled by the ball is
d
= (
v
x
+
u
)
t
= 2
v
y
(
v
x
+
u
)
/g
.
In this time
t
, however, the railcar and boy do not necessarily travel the same distance
as the ball. They travel distance
D
=
ut
+
1
2
at
2
. Thus, the condition that we want is
d
=
D
, or
2
v
y
(
v
x
+
u
)
/g
= 2
v
y
u/g
+
1
2
a
(2
v
y
/g
)
2
.
This simpliﬁes to give
a/g
=
v
x
/v
y
as required.
3.
[5 points.]
A block of mass 2 kg rests on a frictionless platform on wheels. It is
attached to a horizontal spring of spring constant 8 N/m, as shown in the ﬁgure.
Initially the whole system is stationary, but at
t
= 0 the platform begins to move to
the right with a constant acceleration of 2 m/sec
2
. As a result, the block begins to
oscillate horizontally relative to the platform.

2
m/s
2
d
d
2 kg
8 N/m
(a) What would be the new equilibrium extension of the spring?
Imagine that all the
oscillations have died away, but that the platform is still accelerating.
The inertial force on the mass is
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 Fall '07
 Hallet

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