Q3 - Consider an object of mass m rolling down an incline of angle as shown in F

Question

Q3 - Consider an object of mass m rolling down an incline of angle as shown in Figure 3. In terms of known parameters m, g, , , and D, determine the velocity of the object after it has rolled a distance D = d2 - d1 meters down the incline from its stationary initial position. Assume that the object’s potential energy is zero at the height of the detector:

(1) Energy is conserved, Einitial = Efinal, - mgd1sin = - mgd2sin +1/ 2mv2 +1/ 2I2 where I = mR2 is the moment of inertia of the object, = I/mR2 is the constant prefactor determined by the object’s shape, v is the translational velocity of the object, and is the angular velocity.

(2) The rolling objects roll without slipping, = v/R

Q4-A solid sphere and hollow cylinder are rolled down an incline of angle ( = 0.4 and 1 for a solid sphere and hollow cylinder, respectively). What is the ratio of the velocities of the sphere and cylinder, after they have rolled the same distance D down the incline?

Explanation / Answer

Einitial = -mgd1 sin@

Efinal = - mgd2sin@ + m v^2 /2 + Iw^2 /2

     = - m g d2 sin@ + m v^2 /2 + (n m R^2)(v/R)^2 / 2

     = - m g d2 sin@ + mv^2 /2 + n m v^2 /2

Applying energy conservation,

Einitial = Efinal

-mgd1 sin@ = - m g d2 sin@ + mv^2 /2 + n m v^2 /2

m g sin@ ( d1 - d1)   = m v^2 ( 1 + n)/2

g sin@ D = v^2 (n + 1) /2

v= sqrt[ 2g Dsin@ / (n + 1) ]

2) v1 / v2 = sqrt[ 2g Dsin@ / (n1 + 1) ] / sqrt[ 2g Dsin@ / (n2 + 1) ]

everything is same except n.

v1 /v2 = sqrt[ (n2 + 1) / (n1 + 1) ]

n1 = 0.4

cylinder = 1

v1 / v2 = sqrt[ (1 + 1) / (0.4 + 1) ] = 1.195

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