Curvilinear Dynamics and forces

[Sian R]

Homework Statement

A wedge with face inclined at an angle θ to the horizontal is fixed to a rotating turntable. A block of mass m rests on the inclined plane and the coefficient of static friction between the block and the wedge is µ. The block is to remain at position R from the centre of rotation of the turntable.

Draw the forces acting on the block in a diagram and express them in curvilinear coordinates. Show that the acceleration vector of the block is given by a = Rω2nˆ, where ω is the angular velocity of the turntable.

2. Homework Equations

The Attempt at a Solution

I have resolved the forces on the block, there are 4 I think. Friction, which I have said acts in the positive r^ direction, the Normal force from the wedge which is in the ∅^ direction.
Weight I have as W=mg(-sinαr^-cosα ∅^)
centripetal force =IFI(-cosαr^+sinα ∅^)From this and the acceleration equation I have split it into r and ∅ but I am unsure how to prove
a = Rω2nˆ

When I asked my lecturer on how to switch between polar cylindrical into tangential coordinates she said "r-hat points in opposite direction to n-hat, and if the particle moves counterclockwise then t-hat equals
theta-hat"

 

[kuruman]

Judging from the relevant equation you posted, I assume you are using standard cylindrical coordinates where the z-axis is in the same direction as $\vec \omega$. If so, the weight must be in the negative $\hat z$ direction. If that's not the coordinate system you are using, please state what it is that you are using. Also, you missed the force of friction exerted by the plane and you have a "centripetal force". Where did that come from? Only the Earth and the incline can exert forces on the block.

Just draw a free body diagram in cartesian coordinates first. Then re-express Newton's 2nd law in curvilinear coordinates.

 

[Sian R]

Judging from the relevant equation you posted, I assume you are using standard cylindrical coordinates where the z-axis is in the same direction as $\vec \omega$. If so, the weight must be in the negative $\hat z$ direction. If that's not the coordinate system you are using, please state what it is that you are using. Also, you missed the force of friction exerted by the plane and you have a "centripetal force". Where did that come from? Only the Earth and the incline can exert forces on the block.

Just draw a free body diagram in cartesian coordinates first. Then re-express Newton's 2nd law in curvilinear coordinates.

I can’t really explain what I’ve done in writing so I’ve attached my working below, I thought would get a force towards the centre because it is rotating?

 

[kuruman]

I thought would get a force towards the centre because it is rotating?

You should get the net force towards the center. This means that when you add the forces contributed by the incline and gravity, the resultant is a vector that points towards the center. It's called "centripetal" because it is directed towards the center, but that's just a name. You should not use the frame that you show in the picture with one axis along the plane and one perpendicular to it; that will make the analysis unnecessarily complicated. Also, consider that unit vector $\hat r$ is in the horizontal direction radially out, $\hat \theta$ is horizontal and tangent to the circle described by the block (in the same direction as the instantaneous velocity) and $\hat z$ is in the same direction as $\vec \omega$. The unit vectors shown in your drawing are incorrectly identified. I am not sure what the unit vector $\hat n$ mentioned in the statement of the problem is. Is there a definition for it somewhere? Also, the relevant equation that you show looks like an acceleration on the right side and a position on the left side. Where did you get it?

 

[haruspex]

Show that the acceleration vector of the block is given by a = Rω2nˆ, where ω is the angular velocity of the turntable.

Is this the whole of the task? It has nothing to do with all the stuff about the angled wedge and the friction. It is merely asking you to show that the acceleration of a body moving at constant speed in a circle is a radially inward acceleration of rω².

 

[kuruman]

Is this the whole of the task? It has nothing to do with all the stuff about the angled wedge and the friction. It is merely asking you to show that the acceleration of a body moving at constant speed in a circle is a radially inward acceleration of rω².

I believe the first part of the task is

Draw the forces acting on the block in a diagram and express them in curvilinear coordinates.

It is not entirely clear to me how much one has to do to complete the second part

Show that the acceleration vector of the block is given by a = Rω2nˆ, where ω is the angular velocity of the turntable.

 

[Chestermiller]

Show that the acceleration vector of the block is given by a = Rω2nˆ, where ω is the angular velocity of the turntable.

In this equation, what is n^?

 

[kuruman]

In this equation, what is n^?

I have no idea. I expressed my puzzlement about that in post #4, but no answer so far.

 

[haruspex]

In this equation, what is n^?

It ($\hat n$) appears to be the same as $-\hat r$.

 

[Chestermiller]

It ($\hat n$) appears to be the same as $-\hat r$.

I hope so.