A slender rod of constant density lies along the line segment r(t) = tj + (2 - 2t)k, 0≤ t ≤ 1, in the yz-plane. Find the moments of inertia of the rod about the three coordinate axes.

Q: A fluid has density 1200 kg/m3 and flows with velocity ī measured in meters, and the components of ū…

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Q: Let R(t) = et+e be a particle position moving in the plane Then the acceleration at time zero is

A: We have the position vector given in terms of time t.                  Rt=et i+e-t j To find the…

Q: A fluid has density 1200 kg/m³ and flows with velocity ū = xi + yj + zk, where x, y, and z are…

A: Use the integral formula of flow rate given by stoke's theorem to calculate its value. Determine the…

Q: Find the line integral with respect to arc length (8x + 9y)ds, where C is the line segment in the…

A:

Q: For the vector A = ar r2 + az 2z, satisfy the divergence theorem in the circular cylindrical region…

A: Consider the given vector: A→=arr2+az2z and a circular cylindrical region surrounded by  r=5,z=0,…

Q: A fluid has density 1000 kg/m³ and flows with velocity = ci +yj + zk, where x, y, and z are measured…

A: We have to find the rate flow outward through the part of paraboloid . We can to use the methods of…

Q: Find the center of mass and the moment of inertia about the x-axis of a thin rectangular plate…

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Q: Let R(t) = e'i+ ej be a particle position moving in the plane. Then the acceleration at time zero is

A: Given, The position of moving particle is R(t)=et i+e-t j.

Q: For the vector A = ar r2 + az 2z r = 5, z = 0 and z = 4 in the circular cylindrical region…

A: The objective is to prove the divergence theorem for the vector function A=arr2+az2z for the…

Q: Determine the acceleration (vector and magnitude) of the fluid particle occupying the point (-2, 1,…

A: Accleration is given by  a→ =DV→Dt= ∂V→∂t+(V.→∇)V→Here, V→ =2xy i^+xz j^+yz k^Hence, ∂V→∂t= ∂2xy…

Q: A fluid has density 1000 kg/m3 and flows with velocity ū = xi + yj + zk, where x, y, and z are…

A: A fluid density 1000 kg/m3 andflows with velocity v→=xi→+yj→+zk→where x,y and z are measured…

Q: determine the magnitude of the force exerted by block Y on block X during this acceleration.

A: In this question, we need to find the force exerted by block Y on block X.

Q: The velocity field for a fluid is given by V=(X2t3)i+(2XYT)j. The convective acceleration in the…

A: Given, V=x2t3i^+2xytj^ Here, u=x2t3 and v=2xyt Convective acceleration in y-direction is given by:…

Q: Use Stokes' Theorem to caleulate ſ. F - dř, where F = zyi+3=3+5zk, and C is the closed path around a…

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Q: A solid cube, 2 units on a side, is bounded by the planes x = +-1, z = +-1, y = 3, and y = 5. Find…

A: Given:

Q: A solid “trough” of constant density is bounded below by the surface z = 4y2, above by the plane z =…

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Q: Use the Divergence Theorem to find the rate at which a fluid with flux F = through the cube formed…

A: We are asked to find the rate at which a fluid with flux F→=x2+exy, y2+exz, z2+exy through the cube…

Q: The magnitude of the x-component of a unit vector at the point (1, 1) that is normal to $(r) : equi-…

A: The formula to calculate the vector which is normal to the equipotential  line of the potential…

Q: A fluid has density 800 kg/m³ and flows with velocity i = ri + yj+ zk, where x, y, and z are…

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Q: Find the moment of inertia about the z-axis of the surface of the cube cut from the first octant by…

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Q: A particle moves in the xy−plane, so that its coordinates at time t≥0 are x(t)=3t3−3t2, y(t)=20t2+2t…

A: Given

Q: use a parametrization to find the flux ∬S F . n dδacross the surface in the specified direction. F =…

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Q: What would be the resultant force F of the triangular distributed load and its location 'd' measured…

A: Given,

Q: Use Divergence theorem to find the outward flux of F = 2xz i - 2xy j – z2 k across the boundary of…

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Q: What force is required so that a particle of mass m has the position function r(t) = ti + 3t²j +°k?…

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Q: Compute the flux of F= xi+yj+ z*k across the cone z =Vx2 + y? , 0szs1, in the downward direction.…

A: We have to find the flux in the downward direction.   Let us assume that x=rcosθ, y=rsinθ, z=r Then…

Q: Compute the directional derivative of f(r.y) = at the point (1, 1) in a direction normal to the…

A: Given that: f(x,y)=x2y at the point (1,1) Ellipse: x2+2y2=1 at the point (1,0)

Q: Find the line integral with respect to arc length / Q = (0,3). (2x + 7y)ds, where C is the line…

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Q: Determine the acceleration (vector and magnitude) of the fluid particle occupying the point (-2, 1,…

A: The acceleration vector is,                              a =ax + ay +az  where…

Q: Exercise (2): Calculate fe xydx + 3y*dy where C is the curve in xy plane, x = y² from (0,0). to…

A: we have to following integral

Q: Find the mass of a thin wire lyingalong the curve r(t) = √2t i + √2t j + (4 - t2)k, 0 ≤ t ≤ 1,if the…

A: Given curve is  2t i+2t j+(4-t2)k  ,     0≤t≤1   (a) δ=3t (a) δ=1

Q: 4. Compute |F- dš, where F = (y, –x,2z) and S is the surface of the sphere of radius 2, centered at…

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Q: Find the line integral with respect to arc length ∫C(6x+4y)ds, where C is the line segment in the…

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Q: Find the moment of inertia about the -axis of the solidcylinder x 2 +  y 2 < a2 0 < z < h

A: Given:

Q: Express [[[vdV in cylindrical coordinates, where 2 is the solid that lies between cylinders x+y = 1…

A: Region is between the cylinders  x2+y2=1x2+y2=4  Thus,  1≤x≤2  1≤x≤2    Thus, changing to…

Q: A projectile with mass m is launched into the air on a parabolic trajectory. For t >0, its…

A: In the absence of drag or viscous force , the total Mechanical energy of projectile remain constant

Q: A curve in the xy-plane is defined by the vector valued function f(t)= (x(t), y(t)), d'y in terms of…

A: Given : x(t)=2ln(t)y(t)=2sin(-5t)   To find : d2ydx2

Q: Consider the unsteady flow u= u0, v = kt, w = 0 where uo and k are positive constants. Show that the…

A:

Q: use divergence theorem to find the outward flux of f =2xzi-3xyj-z^2k across the boundary of the…

A: By divergence theorem

Q: Find the line integral with respect to arc length Is (3x + 5y)ds, where C is the line segment in the…

A:

Q: Find the moment of mass from the x-axis for y = x In(x) – 3x and y density is equal to 0.37…

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Q: Find the average height of the (single) 34. Average height of a cone cone z = Vx² + y² above the…

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Q: Use Divergence theorem to find the ouward flux of F = 2xz i - 3xy j – z2 k across the boundary of…

A:

Q: A fluid has density 600 kg/m3 and flows with velocity ū = ri + y. measured in meters, and the…

A: Here we have, Velocity v→=xi^+yj^+zk^ where x,y, and z are measured in meters.  the equation of…

Q: A fluid has density 1200 kg/m³ and flows with velocity v = xi + yj + zk, where x, y, and z are…

A:

Q: Use Divergence theorem to find the ouward flux of F = 2xz i - 3xy j – z2 k across the boundary of…

A:

Q: Let I be the moment of inertia of the plate with density p = 1 bounded by y = ez, y = e, x = 0 about…

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Q: Suppose n is a vector normal to the tangent plane of the surface F(x, y, z) = 0 at a point. How is n…

A: We have asked how a vector n which is normal to the tangent plane of the surface F(x,y,z)=0 at a…

Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.