Rolling without slipping & linear acceleration vector

[Joa Boaz]

Homework Statement

Rolling without slipping

A) Derive the linear acceleration vector equations for points A, B, C, and O in terms of R, ω, α and θ at this instant.

B) R = 0.5 m, ω=-54 r/s and α = 0. Determine the MPH of the vehicle and the vector accelerations of points A, B, C, and O.

C) R = 0.5 m, ω=-54 r/s and α = +4.9 rad/sec/sec. Find the magnitude and direction of the acceleration of points O & C.

D) R = 0.5 m, ω=-54 r/s and α = +4.9 rad/sec/sec. Determine the magnitude and direction of the acceleration of the vehicle in "g" units and how long it would take to stop.

Homework Equations

The Attempt at a Solution

A)
A_A = R α j + R_c (-ω² R) i
A_B = 2 R α i + ω² R j
A_O = R α i
A_C = ω² R j

B)
V = R ω
V = 0.5 x 54
= 27 m/s
= 60.4 mph

A_A = -ω² R = -1458 i m/s²
A_B = - 1458 j
A_O = 0
A_C = 1458 j

C)
At O A_O = R α i ⇒ 0.5 x 4.9 = 2.45 i m/s²
At C A_C = 1458 j

D)
A = R α
= 2.45 m/s²

A = 0.25 g

t= ω/α
t= 11.02 sThe above was my attempt. Am I on the right track? Or am I doing this wrong?

 

[haruspex]

I disagree with several of your answers in part A. (What is R_C?) Haven't looked beyond that.
I think the easiest way is to write a vector expression for the location of a point on the circumference in terms of the location of the centre and an angle theta (i.e. theta =0 would be point A, etc.) then differentiate as necessary.

 

[Joa Boaz]

Thank you.

So, if i wrote a general velocity vector equation for any point on the periphery of the wheel

V_p = V_o + ω X R_op

By differentiating so
A_A = A_C + ω × (ω × R_AC) + α × R_AC

 

[haruspex]

Thank you.

So, if i wrote a general velocity vector equation for any point on the periphery of the wheel

V_p = V_o + ω X R_op

By differentiating so
A_A = A_C + ω × (ω × R_AC) + α × R_AC

Close, but I think you have some sign errors. It's a bit confusing because if omega is positive then v₀ is negative, etc.

 

[Joa Boaz]

Close, but I think you have some sign errors. It's a bit confusing because if omega is positive then v₀ is negative, etc.

Thank

I am sorry but why would a positive omega means a negative V_O ?

 

[haruspex]

Thank

I am sorry but why would a positive omega means a negative V_O ?

According to the diagram, theta is measured anticlockwise, so a positive omega would mean rolling to the left. The x-axis is positive to the right.