45 Degrees Does Not Halve Gravitational Field Intensity ```Name: Kyler Status: student Grade: 9-12 Location: AZ Country: USA Date: N/A ``` Question: 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? Replies: Hi Kyler 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. Bob Froehlich 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. -- Tim Mooney 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 Instructor B.S. Earth Sciences; M.S. Geophysics Oklahoma State Univ. Inst. of Technology Kyler, 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 Physics Instructor Illinois Central College Click here to return to the Mathematics Archives

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