A small block with mass m is placed inside an inverted cone that is rotating about a vertical axis such that the time for one revolution of the cone is T (Figure 1). The walls of the cone make an angle β with the horizontal. The coefficient of static friction between the block and the cone is μs . If the block is to remain at a constant height h above the apex of the cone, what are the maximum value of T and the minimum value of T? (That is, find expressions for Tmax and Tmin in terms of β and h.) (I) Although we want the block not to slide up or down on the inside of the cone, this is not an equilibrium problem. The block rotates with the cone and is in uniform circular motion, so it has an acceleration directed toward the center of its circular path. Identify the forces on the block. Select all that apply, from the list below: a. the friction force b.the weight of the block c.the weight of the cone d. the tension force e. the normal force (III) What is the direction of the friction force when the cone is rotating as slowly as possible, so T has its maximum value Tmax? (Select one option from the list below) a.up along the direction of cone's inner surface b. down along the direction of cone's inner surface c. clockwise d. horizontally towards the rotational axis at the cone's center e. counterclockwise (IV)What is the direction of the friction force when the cone is rotating as rapidly as possible, so T has its minimum value Tmin? (Select one option from the list below) a.up along the direction of cone's inner surface b. down along the direction of cone's inner surface c. clockwise d. horizontally towards the rotational axis at the cone's center e. counterclockwise When T=TmaxT=Tmax and T=TminT=Tmin does the static friction force have its maximum magnitude? Why or why not? (Pick one option below) a. When T=Tmax, the radial acceleration is maximal, so the normal force exerted on the block, and hence, the static friction force has its maximum magnitude. When T=Tmin, the static friction force has its minimum. b. The static friction force has its maximum magnitude in both cases, when T=Tmax and T=Tmin .The only difference is in the direction of the friction force. c. When T=Tmin, the radial acceleration is maximal, so the normal force exerted on the block, and hence, the static friction force has its maximum magnitude. When T=Tmax, the static friction force has its minimum. d. The static friction force has its minimum magnitude in both cases, when T=Tmax and T=Tmin. The only difference is in the direction of the friction force.
A small block with mass m is placed inside an inverted cone that is rotating about a vertical axis such that the time for one revolution of the cone is T (Figure 1). The walls of the cone make an angle β with the horizontal. The coefficient of static friction between the block and the cone is μs . If the block is to remain at a constant height h above the apex of the cone, what are the maximum value of T and the minimum value of T? (That is, find expressions for Tmax and Tmin in terms of β and h.)
(I) Although we want the block not to slide up or down on the inside of the cone, this is not an equilibrium problem. The block rotates with the cone and is in uniform circular motion, so it has an acceleration directed toward the center of its circular path.
Identify the forces on the block.
Select all that apply, from the list below:
a. the friction force
b.the weight of the block
c.the weight of the cone
d. the tension force
e. the normal force
(III) What is the direction of the friction force when the cone is rotating as slowly as possible, so T has its maximum value Tmax? (Select one option from the list below)
a.up along the direction of cone's inner surface
b. down along the direction of cone's inner surface
c. clockwise
d. horizontally towards the rotational axis at the cone's center
e. counterclockwise
(IV)What is the direction of the friction force when the cone is rotating as rapidly as possible, so T has its minimum value Tmin? (Select one option from the list below)
a.up along the direction of cone's inner surface
b. down along the direction of cone's inner surface
c. clockwise
d. horizontally towards the rotational axis at the cone's center
e. counterclockwise
When T=TmaxT=Tmax and T=TminT=Tmin does the static friction force have its maximum magnitude? Why or why not? (Pick one option below)
a. When T=Tmax, the radial acceleration is maximal, so the normal force exerted on the block, and hence, the static friction force has its maximum magnitude. When T=Tmin, the static friction force has its minimum.
b. The static friction force has its maximum magnitude in both cases, when T=Tmax and T=Tmin .The only difference is in the direction of the friction force.
c. When T=Tmin, the radial acceleration is maximal, so the normal force exerted on the block, and hence, the static friction force has its maximum magnitude. When T=Tmax, the static friction force has its minimum.
d. The static friction force has its minimum magnitude in both cases, when T=Tmax and T=Tmin. The only difference is in the direction of the friction force.
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