Rolling without slipping & linear acceleration vector

In summary: If the wheel is rolling to the left, then the point at the top (A) is moving to the left. If theta=0 is point A, then the x and y coordinates of A are both negative. The x component of the velocity is just omega times the y position of A. The y component of the velocity is -omega times the x position of A. So the velocity vector is negative in both x and y directions. That's why I said the signs are confusing. The actual velocity vector is positive because it is pointing to the left (negative x) and down (negative y) but the components are negative.
  • #1
Joa Boaz
14
1

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.

media-f53-f537a8aa-70c1-4eab-86cd-d6b687139ba8-phpuy2kkq.png


Homework Equations

The Attempt at a Solution


A)
AA = R α j + Rc (-ω2 R) i
AB = 2 R α i + ω2 R j
AO = R α i
AC = ω2 R j

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

AA = -ω2 R = -1458 i m/s2
AB = - 1458 j
AO = 0
AC = 1458 j

C)
At O AO = R α i ⇒ 0.5 x 4.9 = 2.45 i m/s2
At C AC = 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?
 
Physics news on Phys.org
  • #2
I disagree with several of your answers in part A. (What is RC?) 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.
 
  • #3
Thank you.

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

Vp = Vo + ω X Rop

By differentiating so
AA = AC + ω × (ω × RAC) + α × RAC
 
  • #4
Joa Boaz said:
Thank you.

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

Vp = Vo + ω X Rop

By differentiating so
AA = AC + ω × (ω × RAC) + α × RAC
Close, but I think you have some sign errors. It's a bit confusing because if omega is positive then v0 is negative, etc.
 
  • #5
haruspex said:
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 VO ?
 
  • #6
Joa Boaz said:
Thank

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

1. What is rolling without slipping?

Rolling without slipping is a type of motion in which a body or object moves forward while also rotating without any slipping or sliding. This means that the point of contact between the object and the surface it is rolling on is always stationary.

2. What is the difference between rolling with and without slipping?

The main difference between rolling with and without slipping is the presence or absence of sliding motion between the object and the surface it is rolling on. In rolling with slipping, there is sliding motion and the point of contact between the object and the surface is not stationary, while in rolling without slipping, there is no sliding motion and the point of contact remains stationary.

3. How is linear acceleration vector related to rolling without slipping?

In rolling without slipping, the linear acceleration vector is equal to the angular acceleration of the object multiplied by the radius of the object. This means that the linear acceleration of the object is directly proportional to its angular acceleration and the distance from the center of rotation.

4. What is the significance of the coefficient of rolling friction in rolling without slipping?

The coefficient of rolling friction is a measure of the resistance to rolling motion between the object and the surface it is rolling on. In rolling without slipping, the coefficient of rolling friction plays a crucial role in determining whether the object will roll without slipping or if slipping will occur.

5. How does the shape of the object affect rolling without slipping?

The shape of the object can affect rolling without slipping in several ways. Objects with a larger radius will have a smaller linear acceleration for the same angular acceleration, while objects with a smaller radius will have a larger linear acceleration. Additionally, the shape of the object can also affect the coefficient of rolling friction, which can impact the ability of the object to roll without slipping.

Similar threads

Centripetal acceleration and tangential velocity of an object that revolves and rotates
    • Introductory Physics Homework Help
Replies
12
Views
628
Rolling without slipping down an inclined plane
    • Introductory Physics Homework Help
Replies
18
Views
3K
Wooden sphere rolling on a double metal track
    • Introductory Physics Homework Help
3
Replies
97
Views
3K
A sphere rolling without slipping down a hemisphere
    • Introductory Physics Homework Help
Replies
17
Views
2K
Find the force P such that the disk rolls without slip
    • Introductory Physics Homework Help
Replies
1
Views
1K
Rolling without slipping magnitude
    • Introductory Physics Homework Help
Replies
7
Views
2K
Calculus angular acceleration with respect to theta
    • Introductory Physics Homework Help
Replies
2
Views
2K
Cylinders rolling down without slipping
    • Introductory Physics Homework Help
Replies
8
Views
2K
How Fast Can a Ring with Uneven Mass Distribution Roll Without Hopping?
    • Introductory Physics Homework Help
Replies
4
Views
972
What Is the Effect of Friction on the Height of a Sphere Rolling Down a Ramp?
    • Introductory Physics Homework Help
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top