How Does the Angle θ Relate to Φ in Static Equilibrium of a Tilted Rod?

AI Thread Summary
The discussion revolves around determining the relationship between the angles θ and ϕ for a uniform rod in static equilibrium, supported by two strings. The rod is tilted at an angle θ, while the string attached to the ceiling makes an angle ϕ with the vertical. The participant outlines their approach, considering three forces: the tensions in the strings and the gravitational force at the center of mass. They initially struggle with the torque equations but eventually identify a mistake in their angle calculations, which leads to a corrected equation involving trigonometric functions. The conversation highlights the complexities of static equilibrium problems and the importance of accurately applying trigonometric relationships.
NJJ289
Messages
18
Reaction score
0

Homework Statement


11-p-034.gif



The uniform rod in the figure is supported by two strings. The string attached to the wall is horizontal, and the string attached to the ceiling makes an angle of ϕ with respect to the vertical. The rod itself is tilted from the vertical by an angle θ. If ϕ = 29.9°, what is the value of θ?


Homework Equations



Tcc=Tcw


The Attempt at a Solution



I've approached the problem with three forces acting on the object. the two string tensions at the ends of the object, and a gravity force acting on its center of mass.

The center of mass is considered the location of the axis. This let's the gravity force= zero torque and reduces the problem to simply setting the tension-torques equal since the object is in equilibrium.

so I tried

T1rsin(a)=T2rsin(b)

[r is distance to center of mass from end, and a and b are angles between force vectors and position vector r. t1 and t2 are tension forces]

this simplified out to

sin(90+\theta)cos(90-\phi)=sin(90+\phi)... which does not yeild an answer.

I subbed T1=T2cos(90-\theta) to do this since T1 and T2 have cancelling horizontal forces.

I have literally spent hours on this problem and I'm completely stumped. Any assistance is greatly appreciated!
 
Physics news on Phys.org
Yeash! got it... made a small mistake with the angles.

it's cos(90-phi)sin(90+theta)=sin(180-phi+theta) in case anyone was wondering
 
Kindly see the attached pdf. My attempt to solve it, is in it. I'm wondering if my solution is right. My idea is this: At any point of time, the ball may be assumed to be at an incline which is at an angle of θ(kindly see both the pics in the pdf file). The value of θ will continuously change and so will the value of friction. I'm not able to figure out, why my solution is wrong, if it is wrong .
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...
Back
Top