Equilibrium of Uniform Sphere on Step: Examining the Force P

AI Thread Summary
A uniform sphere of radius a and weight W is analyzed as it rests on a step of height a/5, with friction coefficients of 3/4 between the sphere and ground, and µ between the sphere and step. The equilibrium conditions indicate that the sphere will rotate about the step when the applied force P reaches W/3. Conversely, slipping occurs when P equals 1/6W(3+µ)/(2-µ). The discussion emphasizes the importance of calculating moments and balancing forces to determine the conditions for slipping versus rotation. The range of values for µ that lead to slipping is also a key focus, highlighting the relationship between friction and stability in this scenario.
Johnny87
Messages
9
Reaction score
0
Uniform sphere of radius a and weight W is resting on a horizontal ground in contact with a step of height a/5.

coefficient of friction between sphere and the ground is 3/4
coefficient of friction between sphere and the step is µ

A gradually increasing force P is applied to the highest point of the sphere in a direction perpendicular to the edge of the step.

If equilibrium is broken by the sphere rotating about the step (rather than by slipping against the step and the ground) show that this happens when P=W/3.

If, on the other hand, equilibrium is broken by slipping, show that this happens when P=1/6W(3+µ)/(2-µ)

For what range of values of µ is the equilibrium broken by slipping?




Have no clue at all.
 
Physics news on Phys.org
Write down what clues you do have about friction and the forces in equilibrium in general..
 
Ok, I've tried like this:

To find P rotating:
I took the moment about the centre of the sphere and obtained:
P=mu Reaction(step)

To find P slipping:
Balance:
Horiz.: P + mu (4/5) Reaction(step) + (3/4) Reaction(ground) = (3/5) Reaction(step)
Vertical: W = Reaction(ground) + (4/5) Reaction(step) + mu (3/5) Reaction(step)
(Should I take also the moment? If yes, about which point?)

Then I sketched the graph, and equated the two pulls to see where they intersecate (to find the range of mu) but the result wasn't the one I expected.

What's wrong with my reasoning? Please help me!
 
Draw a diagram first. Suppose the circle touches the ground at G, and intersects the edge of the step at E. Notice that the vertical side of the step is not a tangent to the circle. It’s best to take moment about all the forces about E. We have to find the horizontal dist between the E and the line of action of W. (You calculate it to verify.) When P is max and the sphere is just about to rotate about E, then,

P(a+4a/5) = W*3a/5, where 3a/5 is the horizontal dist between G and the bottom of the step => P = W/3.

Can you do the next part?
 
I'll try. I'm supposed to take moments and balance, right?
(Sorry for these very stupid questions, but I'm not very good at statics..)
 
That's right. Also, since this is statics, the net force on the body is zero. So, the sum of the horizontal forces and the sum of the vertical forces must individually be zero. Actually, it's valid for the sum in any direction, but generally it's found to be convenient to resolve the forces horizontally or vertically.
 

Thread 'Struggling to understand the concept of this question (ice cube in water optics question)'

TL;DR Summary: I came across this question from a Sri Lankan A-level textbook. Question - An ice cube with a length of 10 cm is immersed in water at 0 °C. An observer observes the ice cube from the water, and it seems to be 7.75 cm long. If the refractive index of water is 4/3, find the height of the ice cube immersed in the water. I could not understand how the apparent height of the ice cube in the water depends on the height of the ice cube immersed in the water. Does anyone have an...
View full post »
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
View full post »

Similar threads

Static equilibrium | Uniform sphere on incline | Determining friction
Replies
43
Views
2K
To calculate the forces on a composite body
Replies
2
Views
2K
Ring and solid sphere rolling down an incline - rotation problem
Replies
8
Views
3K
Calculating Forces on a Motorcycle Riding in a Transparent Sphere
Replies
5
Views
2K
How High Should the String Be Above the Floor to Keep the Sphere Sliding?
Replies
8
Views
2K
Sphere striking an incline (not asking for solutions....)
Replies
3
Views
1K
Tension in a deformed cylindrical bag on a surface
Replies
18
Views
1K
Normal force at the base of a ladder
Replies
20
Views
2K
How Is Torque Equilibrium Applied to a Sphere Tethered to a Wall?
Replies
5
Views
6K
Equilibrium of a system consisting of two bodies
Replies
9
Views
2K

Hot Threads

Recent Insights

Back
Top