The assembly consists of two blocks A and B, which have a mass of 20 kg and 30 kg, respectively. Determine the distance B must d
escend in order for A to achieve a speed of 3 m>s starting from rest.
2 answers:
Answer:
ΔδB = 5.7 m
Explanation:
Given
mA = 20 Kg
mB = 30 Kg
vA = 3 m/s
vA0 = vB0 = 0 m/s (the blocks released from rest)
Since the cable length (L) is constant
3*δA + δB = L
3*ΔδA = - ΔδB
If we apply
d(3*δA + δB)/dt = dL/dt
3*vA + vB = 0 ⇒ 3*vA = - vB
vB = -3*(3 m/s) ⇒ vB = - 9 m/s
We apply Conservation of Energy
TA0 + TB0 + vA0 + vB0 = TA + TB + vA + vB
0 + 0 + 0 + 0 = 0.5*mA*vA² + 0.5*mB*vB² + WA*ΔδA + WB*ΔδB
0 = 0.5*mA*vA² + 0.5*mB*(-3*vA)² + WA*(ΔδB/3) - WB*ΔδB
0 = 0.5*20 Kg*(3 m/s)² + 0.5*30 Kg*(-9 m/s)² + (20 Kg*9.81 m/s²)*(ΔδB/3) - (30 Kg*9.81 m/s²)*ΔδB
⇒ ΔδB = 5.7 m
The pic shown can help us to understand the question.
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Answer:
Explanation:
In an equilateral trinagle the center of mass is at 1/3 of the height and horizontally centered.
We can consider that the weigth applies a torque of T = W*b/2 on the right corner, being W the weight and b the base of the triangle.
The weigth depends on the size and specific gravity.
W = 1/2 * b * h * L * SG
Then
Teq = 1/2 * b * h * L * SG * b / 2
Teq = 1/4 * b^2 * h * L * SG
The water would apply a torque of elements of pressure integrated over the area and multiplied by the height at which they are apllied:
The term sin(30) is because of the slope of the wall
The pressure of water is:
p(y) = SGw * (h - y)
Then:
T1 = SGw * sin(30) * L * (h^2*y - h*y^2 + 1/3*y^3)(evaluated between 0 and h)
T1 = SGw * sin(30) * L * (h^2*h - h*h^2 + 1/3*h^3)
T1 = SGw * sin(30) * L * (h^3 - h^3 + 1/3*h^3)
T1 = 1/3 * SGw * sin(30) * L * h^3
To remain stable the equilibrant torque (Teq) must be of larger magnitude than the water pressure torque (T1)
1/4 * b^2 * h * L * SG > 1/3 * SGw * sin(30) * L * h^3
In an equilateral triangle h = b * cos(30)
1/4 * b^3 * cos(30) * L * SG > 1/3 * SGw * sin(30) * L * b^3 * (cos(30))^3
SG > SGw * 4/3* sin(30) * (cos(30))^2
SG > 1/2 * SGw
For the dam to hold, it should have a specific gravity of at leas half the specific gravity of water.
This is avergae specific gravity, including holes.