Answer the following questions regarding Investigation 2. Consider an object of

Question

Answer the following questions regarding Investigation 2. Consider an object of mass m rolling down an incline of angle theta as shown in Figure 3. In terms of known parameters m, g, eta, theta and D, determine the velocity of the object after it has rolled a distance D = d_2 - d_1 meters down the incline from its stationary initial position. Assume that the object's potential energy is zero at the height of the detector. Energy is conserved, E_init = E_final where I = eta m R^2 is the moment of inertia of the object, eta = 1/mR^2 is the constant prefactor determined by the object's shape, v is the translational velocity of the object, and omega is the angular velocity. The rolling objects roll without slipping, omega = v/R Figure 4. Experimental parameters for Question 3

Explanation / Answer

Moment of Inertia of Solid sphere, I = 0.4 m*r^2
Moment of Inertia of Hollow Cylinder, I = m*r^2

Initial Potential Energy = Final Potential Energy + Final Kinetic Energy
m*g*d2*sin() = m*g*d1*sin() + 1/2*mv^2 + 1/2*I*w^2
m*g*d2*sin() = m*g*d1*sin() + 1/2*mv^2 + 1/2*n*m*r^2*v^2/r^2
g*d2*sin() = g*d1*sin() + 1/2*v^2 + 1/2*n*v^2

For Sphere,
g*d2*sin() = g*d1*sin() + 1/2*vsphere^2 + 1/2*0.4*vsphere^2
g*sin()*(d2 - d1) = 0.7*vsphere^2
vsphere = sqrt[g*sin()*(d2 - d1)]/0.7

For Cylinder,
g*d2*sin() = g*d1*sin() + 1/2*vcylinder^2 + 1/2*vcylinder^2
vcylinder =  sqrt[g*sin()*(d2 - d1)]

Vsphere/Vcylinder = 1/sqrt(0.7) = 1.2

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