We determine the release height needed to complete a vertical loop without losing contact with the track. The acceleration in circular motion and the equation of motion explain why this height is necessary.
Adjust conditions:
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半径1.0 mのループを、離れずに一周できる最小の出発高さは？
手前の水平区間に摩擦があると、離れずに一周できる最小の出発高さは？
The equations explain the phenomenon
1Position on the circle is expressed mathematicallyr=r (cos ωt, sin ωt)……
2Differentiation gives the velocityv=r ω (−sin ωt, cos ωt)……
3Differentiating again gives the accelerationa=−r ω² (cos ωt, sin ωt)……
4The magnitude of the acceleration followsa=r ω²=v²r……
5The acceleration is substituted into the equation of motionm v²r=F……
6Acceleration is resolved into components when speed varies in circular motionacenter=v²r，atangential=dvdt……
7The condition for maintaining contact at the top of a vertical circle followsv²≧g r……
8Conservation of energy gives the required release heighth≧2.5 r……
Further study covers other examples of circular motion and centrifugal force.
9The result applies to a car rounding a curvev≦√(μ g r)……
10The result applies to a conical pendulumω²=gL cosθ……
11The motion is viewed in a frame rotating with the object0=F−m r ω²……
What this chapter has shown
In uniform circular motion, the changing direction of velocity produces an inward acceleration v²/r. When speed varies, there is also a tangential component dv/dt. In a frame rotating at the same constant angular velocity as the object, the equation of motion includes an outward centrifugal force.
2.50 mMinimum height for completing the loop without losing contact
For an object released from rest on the inside of a frictionless track, maintaining contact throughout the loop requires a release height of at least 2.5r.
In everyday lifeAt the same speed, a sharper curve requires a greater frictional force.
Detailed summary
The acceleration in circular motion is obtained by differentiating position twice with respect to time. Even when the speed is constant, the direction of motion changes, so the acceleration points toward the center and has magnitude v²/r. In a vertical loop where the speed varies, the acceleration toward the center is still v²/r. The condition that the normal force at the top must be at least 0 requires a starting height of at least 2.5r. Centripetal force is simply the name for the net force directed toward the center, arising from familiar forces such as tension, gravity, and frictional force. Centrifugal force is an apparent force that appears when the equation of motion is written in a rotating coordinate system.