It shows an accelerometer, a device for measuring the horizontal acceleration of cars and airplanes. A ball is free to roll on a parabolic rack described by the equation, where both and are in meters. A scale along the bottom is used to measure the ball's horizontal position.
a. Find an expression that allows you to use a measured position to compute the acceleration. (For example, is a possible expression.)
b. What is the acceleration if
(a) An expression that allows to use measured the acceleration is
(b) The acceleration of
(a) Let's draw a free body diagram when the system is accelerating.
It is crucial to state that there is no friction, since a friction force, due to the discontinuity at the maximum , would not allow for a continuous measurement spectrum
Having drawn the free body diagram, we can project to the vertical and horizontal axes and have:
Substituting, we have
The key to solving is realizing that the tangent of our angle is equal to the slope of the line tangent to the urve at the same point. The latter is easily determined, as one remembers from calculus, as
Combining, we have
Please note that we can't perform dimensional analysis on this result.
the acceleration will be
If you are not sure why the angle between and the vertical axis is equal to the angle between the line tangent to the curve and the consider the close-up below.
Since is perpendicular to the surface and the line is tangent to this surface. Therefore we have . On the other hand, as opposite angles. But since they are both in the same right triangle. Therefore,
In the first sketch both angles were denoted by
An accident victim with a broken leg is being placed traction. The patient wears a special boot with a pulley attached to the sole. The foot and boot together have a mass of 4.0 kg and the doctor has decided to hang a 6.0 kg mass from the rope. The boot is held suspended by the ropes, as shown in FIGURE P6.41, and does not touch the bed a. Determine the amount of tension in the rope by using Newton’s laws to analyze the hanging mass. b. The net traction force needs to pull straight out on the leg. What is the proper angle u for the upper rope? c. What is the net traction force pulling on the leg? Hint: If the pulleys are frictionless, which we will assume, the tension in the rope is constant from one end to the other.
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