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A Bug Crawls Towards The Rim With A Constant Speed Vo Along The Spoke Of A Wheel That Is Rotating With Constant Angular Velocity W About A Vertical Axis. Find All Apparent Forces Acting On The Bug. And How Far The Bug Can Crawl Before It Starts To Slip, Given That Coefficient Of Static Friction Between The Bug And The Spoke Is Us.

Q) A Bug Crawls Towards The Rim With A Constant Speed Vo Along The Spoke Of A Wheel That Is Rotating With Constant Angular Velocity W About A Vertical Axis. Find All Apparent Forces Acting On The Bug. And How Far The Bug Can Crawl Before It Starts To Slip, Given That Coefficient Of Static Friction Between The Bug And The Spoke Is Us.



Ans) Apparent Forces Acting on the Bug:

There are three apparent forces acting on the bug:

  1. Centrifugal Force (Fc): This outward force arises due to the bug's circular motion along the rotating wheel.
    • Magnitude: Fc = mv^2/r, where m is the bug's mass, v is its instantaneous velocity relative to the rim (sum of crawling speed and tangential velocity due to wheel rotation), and r is the distance from the bug to the wheel's axis.
    • Direction: Radially outward, away from the wheel's axis.
  2. Normal Force (Fn): This is the force exerted by the spoke on the bug perpendicular to its motion.
    • Magnitude: Fn = N, where N is the reaction force due to the bug pushing against the spoke.
    • Direction: Perpendicular to the spoke, pointing towards the wheel's center.
  3. Frictional Force (Ff): This force opposes the bug's crawling motion due to the friction between the bug and the spoke.
    • Magnitude: Ff = us * Fn, where us is the coefficient of static friction.
    • Direction: Opposed to the bug's crawling direction along the spoke.

Slipping Condition:

The bug will start to slip when the frictional force is not sufficient to counter the tangential component of the centrifugal force. Mathematically, this occurs when:

Ff < mv^2/r * sin(theta)

where theta is the angle between the spoke and the tangential direction (90 degrees at the rim).

Substituting with the expressions for Ff and Fn:

us * N < mv^2/r * sin(theta)

Solving for N (normal force):

N > mv^2/(r * us * sin(theta))

Therefore, the bug can crawl as far as the spoke can provide a normal force greater than the right-hand side of the equation above. This distance depends on the wheel's geometry, the bug's mass and crawling speed, the frictional coefficient, and the angle at which the bug reaches the rim.

Note: This solution assumes the wheel is rotating in a horizontal plane. If the wheel is tilted, additional forces due to gravity and the inclined plane may need to be considered.




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