1. The problem statement, all variables and given/known data
We consinder a doble cone with a radius R and an angle α (pike) and the mass m. It is located on two rails with an opening angle β. The rails enclose the angle γ with the ground. A is the lowest point of the rails.
First, the center of mass of the double cone is locatd vertically above the point A regarding the plane described by the two rails. The double cone ascends the rails. You can assume that base is in the center between the rails and that the line through the center of mass and the contact points with the rails is orthogonal to the rails.
a) Why does the double cone ascends the rails? State the conditions for the angles α,β,γ so that this happen.
b) Proof that the moment of inertia regarding the connecting line through through the pikes is I=3/10*m*R2
c) Determine the function v(d). v is the velocity of the center of mass and d is the rolled distance on the rails.
d) Calculate for the values α=50°; β=40°; γ=5°; R=10cm and m=100g the distance L which the double cone rolls on the rails and the maximal reached velocity.
2. Relevant equations
3. The attempt at a solution
a) The reason is that the height of the center of mass decreases while the double cone is ascending the rails. Condition: tan(γ)<tan(α/2)*tan(β/2)
b) Easy proof with the integral-formula of the moment of inertia.
However, I do not have ideas how to solve c) and d). Can you help me?
We consinder a doble cone with a radius R and an angle α (pike) and the mass m. It is located on two rails with an opening angle β. The rails enclose the angle γ with the ground. A is the lowest point of the rails.
First, the center of mass of the double cone is locatd vertically above the point A regarding the plane described by the two rails. The double cone ascends the rails. You can assume that base is in the center between the rails and that the line through the center of mass and the contact points with the rails is orthogonal to the rails.
a) Why does the double cone ascends the rails? State the conditions for the angles α,β,γ so that this happen.
b) Proof that the moment of inertia regarding the connecting line through through the pikes is I=3/10*m*R2
c) Determine the function v(d). v is the velocity of the center of mass and d is the rolled distance on the rails.
d) Calculate for the values α=50°; β=40°; γ=5°; R=10cm and m=100g the distance L which the double cone rolls on the rails and the maximal reached velocity.
2. Relevant equations
3. The attempt at a solution
a) The reason is that the height of the center of mass decreases while the double cone is ascending the rails. Condition: tan(γ)<tan(α/2)*tan(β/2)
b) Easy proof with the integral-formula of the moment of inertia.
However, I do not have ideas how to solve c) and d). Can you help me?
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