In this problem you will consider the motion of a cylinder of radius rcyl𝑟cyl that is rolled from a certain height hℎ so that it "loops the loop," that is, rolls around the track with a loop of radius rloop𝑟loop shown in the figure without losing contact with the track. (Figure 1)Unless otherwise stated, assume that friction is sufficient that the cylinder rolls without slipping. The radius rcyl𝑟cyl of the cylinder is much smaller than the radius rloop𝑟loop of the loop.Figure1 of 1Part ACompared to an object that does not roll, but instead slides without friction, should a rolling object be released from the same,a greater, or a lesser height in order just barely to complete the loop the loop?View Available Hint(s)for Part AThe rolling object should be released from a greater height.The rolling object should be released from a lesser height.The rolling object should be released from exactly the same height.The answer depends on the moment of inertia of the rolling object.SubmitPart BFind the minimum height hℎ that will allow a solid cylinder of mass m𝑚 and radius rcyl𝑟cyl to loop the loop of radius rloop𝑟loop.Express hℎ in terms of the radius rloop𝑟loop of the loop.View Available Hint(s)for Part B
Question
In this problem you will consider the motion of a cylinder of radius rcyl𝑟cyl that is rolled from a certain height hℎ so that it "loops the loop," that is, rolls around the track with a loop of radius rloop𝑟loop shown in the figure without losing contact with the track. (Figure 1)Unless otherwise stated, assume that friction is sufficient that the cylinder rolls without slipping. The radius rcyl𝑟cyl of the cylinder is much smaller than the radius rloop𝑟loop of the loop.Figure1 of 1Part ACompared to an object that does not roll, but instead slides without friction, should a rolling object be released from the same,a greater, or a lesser height in order just barely to complete the loop the loop?View Available Hint(s)for Part AThe rolling object should be released from a greater height.The rolling object should be released from a lesser height.The rolling object should be released from exactly the same height.The answer depends on the moment of inertia of the rolling object.SubmitPart BFind the minimum height hℎ that will allow a solid cylinder of mass m𝑚 and radius rcyl𝑟cyl to loop the loop of radius rloop𝑟loop.Express hℎ in terms of the radius rloop𝑟loop of the loop.View Available Hint(s)for Part B
Solution
Part A
A rolling object should be released from a greater height. This is because a rolling object has both translational and rotational kinetic energy, whereas a sliding object only has translational kinetic energy. Therefore, a rolling object needs a greater initial potential energy (i.e., a greater height) to complete the loop.
Part B
The minimum height h that will allow a solid cylinder of mass m and radius rcyl to loop the loop of radius rloop can be found by equating the initial potential energy of the cylinder to the sum of its translational and rotational kinetic energy at the top of the loop.
The initial potential energy is mgh, where g is the acceleration due to gravity.
The translational kinetic energy at the top of the loop is (1/2)mv^2, where v is the speed of the center of mass of the cylinder.
The rotational kinetic energy at the top of the loop is (1/2)Iω^2, where I is the moment of inertia of the cylinder and ω is the angular velocity.
Since the cylinder rolls without slipping, v = ωr.
The moment of inertia I of a solid cylinder is (1/2)mr^2.
Substituting these values into the energy equation and solving for h gives:
h = (5/2)rloop
This is the minimum height from which the cylinder must be released in order to complete the loop.
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