A solid ball of mass m and radius r rolls without slipping through a loop of rad
ID: 2231478 • Letter: A
Question
A solid ball of mass m and radius r rolls without slipping through a loop of radius R, as shown in the figure. From what height h should the ball be launched in order to make it through the loop without falling off the track? (Use any variable or symbol stated above along with the following as necessary: g.)
A solid ball of mass m and radius r rolls without slipping through a loop of radius R, as shown in the figure. From what height h should the ball be launched in order to make it through the loop without falling off the track? (Use any variable or symbol stated above along with the following as necessary: g.)Explanation / Answer
m = mass of ball r = radius of ball h = release height R = radius of loop g = acceleration of gravity v = linear velocity of ball w = angular velocity of ball = v/r I = moment of inertia of solid ball = 2/5mr² GPE at launch height (ball at rest) = mgh Total Mech Energy (ball at top of loop): linear KE + rotational KE + GPE = 1/2mv² + 1/2Iw² + mg(2R) by Conservation of Energy: mgh = 1/2mv² + 1/2Iw² + 2mgR mgh = 1/2mv² + 1/2(2/5)mr²(w²) + 2mgR cancel m's gh = 1/2v² +1/5r²(v/r)² + 2gR gh = v²/2 + v²/5 +2gR To remain in loop at top position, centripetal force(Fc) must equal weight of ball (mg) Fc = mg mv²/R = mg v² = Rg substitute in equation with "h" above: gh = Rg/2 + Rg/5 + 2gR cancel g's h = R/2 + R/5 + 2R = 2.7 R ANS
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.