1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
kobusy [5.1K]
3 years ago
13

A torsion-bar spring consists of a prismatic bar, usually of round cross section, that is twisted at one end and held fast at th

e other to form a stiff spring. An engineer needs a stiffer one than usual and so considers building in both ends and applying the torque somewhere in the central portion of the span, as shown in the figure. This effectively creates two springs in parallel.
The engineer is forced by geometric considerations to apply the torque on the spring at the location x = 0.37l. For a uniform-diameter spring, this would cause one leg of the span to be underutilized when both legs have the same diameter. For optimal design the diameter of each leg should be designed such that the maximum shear stress in each leg is the same Using x = 0.37l, l=10 in, T=1,500 lbf.in, and G = 10.4 Mpsi, design the spring such that the maximum shear stresses in each leg are equal and the spring will have the same spring rate (angle of twist) as a uniform-diameter bar with T=1,500 lbf in, x=1/2-5 in, and d1=d2=0.5
Find the required diameters of each leg:
d1
d2
Physics
1 answer:
ra1l [238]3 years ago
7 0

Answer:

d₁ = 0.29 in

d₂ = 0.505 in

Explanation:

Given:

T = 1500 lbf in

L = 10 in

x = 0.5 L = 5 in

T_{1} =\frac{T(L-x)}{L} =\frac{1500*(10-5)}{10} =750lbfin

First case: T = T₁ + T₂

T₂ = T - T₁ = 1500 - 750 = 750 lbf in

If the shafts are in series:

θ = θ₁ + θ₂

θ = ((T₁ * L₁)/GJ) + ((T₂ * L₂)/GJ)

Second case: If d₁ ≠ d₂

θ = ((T₁ * L₁)/GJ₁) + ((T₂ * L₂)/GJ₂) = 0 (eq. 1)

t₁ = t₂

\frac{16T_{1} }{\pi d_{1}^{3}  } =\frac{16T_{2} }{\pi d_{2}^{3}  } (eq. 2)

T₁ + T₂ = 1500 (eq. 3)

θ₁ first case = θ₁ second case

Replacing:

\frac{750*5}{G(\frac{\pi }{32})*0.5^{4}  } =\frac{T_{1}*3.7 }{G(\frac{\pi }{32})*d_{1} ^{4}  }\\T_{1} =16216d_{1} ^{4}

The same way to θ₂:

\frac{750*5}{G(\frac{\pi }{32})*0.5^{4}  } =\frac{T_{2}*6.3 }{G(\frac{\pi }{32})*d_{2} ^{4}  } \\T_{2} =9523.8d_{2} ^{4}

From equation 2, we have:

d₁ = 0.587 * d₂

From equation 3, we have:

d₂ = 0.505 in

d₁ = 0.29 in

You might be interested in
A particle initially located at the origin has an acceleration of vector a = 2.00ĵ m/s2 and an initial velocity of vector v i =
natali 33 [55]

With acceleration

\mathbf a=\left(2.00\dfrac{\rm m}{\mathrm s^2}\right)\,\mathbf j

and initial velocity

\mathbf v(0)=\left(8.00\dfrac{\rm m}{\rm s}\right)\,\mathbf i

the velocity at time <em>t</em> (b) is given by

\mathbf v(t)=\mathbf v(0)+\displaystyle\int_0^t\mathbf a\,\mathrm du

\mathbf v(t)=\left(8.00\dfrac{\rm m}{\rm s}\right)\,\mathbf i+\displaystyle\int_0^t\left(2.00\dfrac{\rm m}{\mathrm s^2}\right)\,\mathbf j\,\mathrm du

\mathbf v(t)=\left(8.00\dfrac{\rm m}{\rm s}\right)\,\mathbf i+\left(2.00\dfrac{\rm m}{\mathrm s^2}\right)u\,\mathbf j\bigg|_{u=0}^{u=t}

\mathbf v(t)=\left(8.00\dfrac{\rm m}{\rm s}\right)\,\mathbf i+\left(2.00\dfrac{\rm m}{\mathrm s^2}\right)t\,\mathbf j

We can get the position at time <em>t</em> (a) by integrating the velocity:

\mathbf x(t)=\mathbf x(0)+\displaystyle\int_0^t\mathbf v(u)\,\mathrm du

The particle starts at the origin, so \mathbf x(0)=\mathbf0.

\mathbf x(t)=\displaystyle\int_0^t\left(8.00\dfrac{\rm m}{\rm s}\right)\,\mathbf i+\left(2.00\dfrac{\rm m}{\mathrm s^2}\right)u\,\mathbf j\,\mathrm du

\mathbf x(t)=\left(\left(8.00\dfrac{\rm m}{\rm s}\right)u\,\mathbf i+\dfrac12\left(2.00\dfrac{\rm m}{\mathrm s^2}\right)u^2\,\mathbf j\right)\bigg|_{u=0}^{u=t}

\mathbf x(t)=\left(8.00\dfrac{\rm m}{\rm s}\right)t\,\mathbf i+\left(1.00\dfrac{\rm m}{\mathrm s^2}\right)t^2\,\mathbf j

Get the coordinates at <em>t</em> = 8.00 s by evaluating \mathbf x(t) at this time:

\mathbf x(8.00\,\mathrm s)=\left(8.00\dfrac{\rm m}{\rm s}\right)(8.00\,\mathrm s)\,\mathbf i+\left(1.00\dfrac{\rm m}{\mathrm s^2}\right)(8.00\,\mathrm s)^2\,\mathbf j

\mathbf x(8.00\,\mathrm s)=(64.0\,\mathrm m)\,\mathbf i+(64.0\,\mathrm m)\,\mathbf j

so the particle is located at (<em>x</em>, <em>y</em>) = (64.0, 64.0).

Get the speed at <em>t</em> = 8.00 s by evaluating \mathbf v(t) at the same time:

\mathbf v(8.00\,\mathrm s)=\left(8.00\dfrac{\rm m}{\rm s}\right)\,\mathbf i+\left(2.00\dfrac{\rm m}{\mathrm s^2}\right)(8.00\,\mathrm s)\,\mathbf j

\mathbf v(8.00\,\mathrm s)=\left(8.00\dfrac{\rm m}{\rm s}\right)\,\mathbf i+\left(16.0\dfrac{\rm m}{\rm s}\right)\,\mathbf j

This is the <em>velocity</em> at <em>t</em> = 8.00 s. Get the <em>speed</em> by computing the magnitude of this vector:

\|\mathbf v(8.00\,\mathrm s)\|=\sqrt{\left(8.00\dfrac{\rm m}{\rm s}\right)^2+\left(16.0\dfrac{\rm m}{\rm s}\right)^2}=8\sqrt5\dfrac{\rm m}{\rm s}\approx17.9\dfrac{\rm m}{\rm s}

5 0
2 years ago
Which is a freshwater region characterized as a large standing body of water? A. lake B. river C. ocean D. stream
Alexus [3.1K]
This is A.) lake. A river is a small amount of water that isn't always fresh water. A stream is too small. And an ocean is made of salt water.
4 0
3 years ago
Read 2 more answers
An object is dropped from a vertical distance of 25.5 m above the ground, and it takes 2.28 sec to fall that distance. A second
DENIUS [597]

Answer:

The second object takes 2.28 s to fall the 25.5 m.

Explanation:

In this case, both objects take the same time to fall, since <em>no vertical velocity is added </em>to any of them.

You can also confirm this by sepparating the second's object movement into its two directions: in the horizontal one, we have <em>linear uniform motion, </em>and in the vertical one, we have <em>free fall, </em>with exactly the same characteristics as for the first object.

4 0
3 years ago
A ball is thrown horizontally from a 19 m -high building with a speed of 2.0 m/s . How far from the base of the building does th
bazaltina [42]
Using the formula: ΔY = V₀y * t + (1/2) * ay * t²

Solve for time and get: 1.968s

Then use: v = d/t in the x-direction and get: d = 3.936
3 0
3 years ago
The rock cycle _____.
Reika [66]
B is the answer you need and i honestly got this question on a middle school test

you must be in different area then me

4 0
3 years ago
Read 2 more answers
Other questions:
  • Discuss the correlation or connection between stars with a higher mass and the amount of fuel they have to work with
    8·1 answer
  • in your own words provide two advantages of using meters as a measurement of length rather than old measurements of length such
    12·1 answer
  • 50 grams of ice cubes at -15°C are used to chill a water at 30°C with mass mH20 = 200 g. Assume that the water is kept in a foam
    11·1 answer
  • Round your answers to one decimal place.this parallel circuit has two resistors at 15 and 40 ohms. what is the total resistance?
    12·2 answers
  • Harry is reading an online summary of the law of reflection. The site states that after light hits a mirror, the angle of reflec
    10·2 answers
  • If a truck weighs 18,000 N and it’s tires are inflated to a pressure of 190 kPa how large is the area of the trucks tires that a
    6·1 answer
  • She knows that a magnesium ion has an electric field that pints away from the ion. What values should she use to complete her ta
    15·1 answer
  • Plant and Snail Gizmo
    7·2 answers
  • Why food cook faster with salt water than cook with pure water​
    14·2 answers
  • Uranus and neptune have methane clouds but jupiter and saturn do not. Which factor explains why?.
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!