- #1

pandatime

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- Homework Statement
- Hanging from the ceiling over a baby bed, well out of baby’s reach, is a string with plastic shapes, as shown here. The string is taut (there is no slack), as shown by the straight segments. Each plastic shape has the same mass m, and they are equally spaced by a distance d, as shown. The angles labeled θ describe the angle formed by the end of the string and the ceiling at each end. The center length of sting is horizontal. The remaining two segments each form an angle with the horizontal, labeled ϕ. Let T1 be the tension in the leftmost section of the string, T2 be the tension in the section adjacent to it, and T3 be the tension in the horizontal segment. (a) Find an equation for the tension in each section of the string in terms of the variables m, g, and θ. (b) Find the angle ϕ in terms of the angle θ. (c) If θ=5.10°, what is the value of ϕ? (d) Find the distance x between the endpoints in terms of d and θ.

- Relevant Equations
- F_g = mg

equations to split vector into component form using cos(theta) and sin(theta)

a*b = |a||b|cos(theta) (to find angles of vectors)

I have actually already partly solved a), as I do get the concept behind how to find tension through making sure that the net force in the x and y direction are zero.

Here are my answers for a)

T1 = T5 = 2mg/sin(theta)

T2 = T4 = mg/sin(phi)

T3 = mg*cot(phi)

The reason I am asking this question here is because I cannot figure out how to write phi in terms of theta, which is what b) asks me to do.

I've tried to rearrange

T1 = 2mg/sin(theta)

T1sin(theta)=2mg

(T1sin(theta))/2 = mg

and then plugging it into my T2/T4 equation

so, T4=(T1sin(theta))/2)/sin(phi)

but then I don't know how to get rid of T1, in the T4 equation, as the question explicitly tells me to write my answers in terms of m, g, and theta.

I looked at the answer key to try to nudge me in the right direction and for some reason the answer is phi = arctan(1/2tan(theta)).

I'm thinking that maybe the T3 equation would play a hand in the answer, because it's my only equation that has tan(theta) in it somewhere.

I don't think I see any angle rules that relate phi to theta either, so I couldn't solve it that way.

That being said, any guidance would be appreciated in finding out how to write phi in terms of theta

Here are my answers for a)

T1 = T5 = 2mg/sin(theta)

T2 = T4 = mg/sin(phi)

T3 = mg*cot(phi)

The reason I am asking this question here is because I cannot figure out how to write phi in terms of theta, which is what b) asks me to do.

I've tried to rearrange

T1 = 2mg/sin(theta)

T1sin(theta)=2mg

(T1sin(theta))/2 = mg

and then plugging it into my T2/T4 equation

so, T4=(T1sin(theta))/2)/sin(phi)

but then I don't know how to get rid of T1, in the T4 equation, as the question explicitly tells me to write my answers in terms of m, g, and theta.

I looked at the answer key to try to nudge me in the right direction and for some reason the answer is phi = arctan(1/2tan(theta)).

I'm thinking that maybe the T3 equation would play a hand in the answer, because it's my only equation that has tan(theta) in it somewhere.

I don't think I see any angle rules that relate phi to theta either, so I couldn't solve it that way.

That being said, any guidance would be appreciated in finding out how to write phi in terms of theta

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