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Two planes xyand yz are passing through a point in a strained material. The normal and shear stresses on xy plane are +80 MPa, -30 MPa respectively and normal and shear stresses on plane yz are +30 MPa and +30 MPa respectively. What is the angle between the planes?
- a)180°
- b)135°
- c)90°
- d)46°
Correct answer is option 'C'. Can you explain this answer?
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The angle between the planes can be determined using the formula:
cosθ = (σxy * σyz + τxy * τyz) / sqrt((σxy^2 + τxy^2) * (σyz^2 + τyz^2))
where σxy and σyz are the normal stresses on the xy and yz planes respectively, and τxy and τyz are the shear stresses on the xy and yz planes respectively.
Given:
σxy = 80 MPa
σyz = 30 MPa
τxy = -30 MPa
τyz = 30 MPa
Plugging in these values into the formula:
cosθ = (80 * 30 + (-30) * 30) / sqrt((80^2 + (-30)^2) * (30^2 + 30^2))
cosθ = (2400 - 900) / sqrt((6400 + 900) * (900 + 900))
cosθ = 1500 / sqrt(7300 * 1800)
cosθ = 1500 / sqrt(13140000)
cosθ ≈ 1500 / 3621.32
cosθ ≈ 0.4135
To find the angle, we can take the inverse cosine of 0.4135:
θ ≈ acos(0.4135)
θ ≈ 65.38 degrees
Therefore, the angle between the planes is approximately 65.38 degrees.
cosθ = (σxy * σyz + τxy * τyz) / sqrt((σxy^2 + τxy^2) * (σyz^2 + τyz^2))
where σxy and σyz are the normal stresses on the xy and yz planes respectively, and τxy and τyz are the shear stresses on the xy and yz planes respectively.
Given:
σxy = 80 MPa
σyz = 30 MPa
τxy = -30 MPa
τyz = 30 MPa
Plugging in these values into the formula:
cosθ = (80 * 30 + (-30) * 30) / sqrt((80^2 + (-30)^2) * (30^2 + 30^2))
cosθ = (2400 - 900) / sqrt((6400 + 900) * (900 + 900))
cosθ = 1500 / sqrt(7300 * 1800)
cosθ = 1500 / sqrt(13140000)
cosθ ≈ 1500 / 3621.32
cosθ ≈ 0.4135
To find the angle, we can take the inverse cosine of 0.4135:
θ ≈ acos(0.4135)
θ ≈ 65.38 degrees
Therefore, the angle between the planes is approximately 65.38 degrees.
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Two planes xyand yz are passing through a point in a strained material. The normal and shear stresses on xy plane are +80 MPa, -30 MPa respectively and normal and shear stresses on plane yz are +30 MPa and +30 MPa respectively. What is the angle between the planes?a)180°b)135°c)90°d)46°Correct answer is option 'C'. Can you explain this answer?
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