45 Degrees Does Not Halve Gravitational Field Intensity
If you were to take a long ramp, about 30 feet long,
angle it at a 45 degrees angle to make a right triangle with 45,
45, and 90 degree angles, and roll the ball down it, would it have
half the acceleration of a ball free falling at an initial rate of
9.8 meters per second per second?
I presume your configuration would be similar to an object on a ramp
such that the ramp is elevated at an angle of 45 degrees from the
horizon. In this case, the acceleration of the ball would be 0.707
times 9.8 m/s/s. The reason is that the component of acceleration
pointing down the ramp comes from the straight down acceleration (9.8
m/s/s) multiplied by the sine of the angle above the horizon. In the
limit when the ramp is at 90 degrees to the horizon (straight up and
down), the acceleration of the ball is the full 9.8 m/s/s. When the
angle is zero, the acceleration is zero. To get half the acceleration
due to gravity, the angle should be 30 degrees, as sin(30) is 0.5. The
length of the ramp does not come into play here. Also please note that
we are assuming that the ball has negligible rotational inertia, which
will further decrease the acceleration down the ramp.
Hope this helps.
The ball would have an acceleration of (9.8 m/s^2)*sin(45 deg.) = 6.93 m/s^2
parallel to the ramp. The vertical component of its acceleration would
be sin(45 deg.) of this, which would be half of 9.8 m/s^2.
No, because some of the energy gained by the ball rolling down must
go into spinning the ball (rotational inertia). That's why all the
problems in your Physics book talk about blocks sliding down
inclined planes. Sliding blocks don't spin so the problem of
rotational inertia is neatly avoided.
R. W. "Bob" Avakian
B.S. Earth Sciences; M.S. Geophysics
Oklahoma State Univ. Inst. of Technology
It would have less acceleration, but not necessarily half. That would
depend on whether the ball were solid or hollow. It would depend on
what is called the moment of inertia, or rotational inertia, of the
ball. A hollow ball of mass M and radius R resists rotation more than a
solid ball of the same mass and size. A very light ball with a great
deal of mass at the ball's center would have free fall acceleration
divided by the square root of two. Whenever dealing with an object that
is rotating, structure of the object becomes important.
Dr. Ken Mellendorf
Illinois Central College
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Update: June 2012