Question

Four round objects of equal mass and radius roll without slipping along a horizontal surface that then bends upward and backward into an arc of half a circle. The objects all have the same linear speed initially. The objects are a hollow cylinder, a solid cylinder, a solid sphere, and a hollow sphere. The objects to go up the arc and exit the are going in the opposite direction they entered without falling off the arc. Now, several trials are run for each object. For each trial, the initial speed of the object is reduced until the object does not make it through the full arc. The speed needed for each object to just make it through the arc is recorded. Which of the following correctly lists the objects in order from fastest to slowest speed needed to make it through the arc?
A. Hollow cylinder, hollow sphere, solid cylinder, solid sphere
B. Hollow cylinder, solid sphere, solid cylinder, hollow sphere
C. Solid sphere, solid cylinder, hollow sphere, hollow cylinder
D. Solid sphere, hollow sphere, solid cylinder, hollow cylinder
E. Hollow sphere, solid cylinder, hollow cylinder, solid sphere

Answer 1

Answer:

A

Explanation:

inertia Equation for All Object is k m R²

inertia Equation of Hollow Cylinder I = m R²

inertia Equation of Solid Cylinder I = ½ m R²

inertia Equation of Hollow Sphere I = ⅔ m R²

inertia Equation of Solid Sphere I = 2/5 m R²

to find the speed needed we can use Mechanical Energy (ME). or Total Energy.

ME = Translation Kinetic Energy + Rotation Kinetic Energi

ME = ½ m v² + ½ I w²

ME = ½ m v² (1 + k)

v² = 2ME / m(1+k)

sinnce all objects have the same mass, we can figure that v² is proportionally with 1/(1+k)

v²(hc) = 1/(1+1) = ½ = 0.5

v = 0.707

v²(sc) = 1/(1+½) = 2/3 = 0.67

v = 0.818

v²(hs) = 1/(1+⅔) = 3/5 = 0.6

v = 0.744

v²(ss) = 1/(1+²/5) = 5/7 = 0.71

v = 0.842

these is the ratio of speed they produce in the end of the arc with initial speed = 0 m/s

so, if we must give speed to the object so that it can reach the end of the arc, the fastest speed is given to hollow cylinder and the slowest speed is given to Solid Sphere

Then the answer is

HC, HS, SC, SS