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Let F(r, y)(2ry2, 22y) be a force field. Suppose that a particle begins its motion at the origin and ends its movement at any point in a plane that is not on the x-axis or the y-axis. Furthermore,...
A force acting on a particle in the xy plane at coordinates (x, y) is given...
A force acting on a particle in the xy plane at coordinates (x, y) is given by vector F = (F_0/r) (y hat i - x hat j), where F_0 is a positive constant and r is the distance of the particle from the origin. Show that the magnitude of this force is F_0. Show that the direction of vector F is perpendicular to vector r = x hat i + y hat j.
9. (8 marks) Let F be an inverse force field given by k F(r,y,z) r13 r,...
9. (8 marks) Let F be an inverse force field given by k F(r,y,z) r13 r, where k is a constant. Find the work done by F on an object as its point of application moves along the -axis from A(1,0, 0) to B(2,0,0)
9. (8 marks) Let F be an inverse force field given by k F(r,y,z) r13 r, where k is a constant. Find the work done by F on an object as its point of application moves...
A particle is constrained to move in the x - y plane. Its position as a...
A particle is constrained to move in the x - y plane. Its position as a function of time is given by x(t) = x_0 sin(w_xt) and y(t) = y_0 cos(w_yt). Find the force vector F on the particle. Under which condition is the force a central force (i.e., points in the vector r direction)? Find the potential energy for the general force that you found in part (a) (i.e., not necessarily central). Show that the particle's energy is conserved.
2. Consider a charged particle in a magnetic field. Let -e = -1.60 x 10-191C) be...
2. Consider a charged particle in a magnetic field. Let -e = -1.60 x 10-191C) be an electron, under B=0.1T) of a uniform magnetic field directed out of the page direction (o-direction). At t = 0[sec] the particle is at the origin, and the initial velocity is toward positive y direction. (a) Sketch the initial configuration of this electron (-e at the origin) into (x,y)-plane. Applying the right hand rule, indicate the magnetic force on this moving electron. Note that...
. Let F(a,y)-(3e +secztan,e -90 (a) Show that F is a conservative. (b) Find a function f (potenti...
. Let F(a,y)-(3e +secztan,e -90 (a) Show that F is a conservative. (b) Find a function f (potential function) show that F (c) Use above result to evaluate Jc F. V. dr, where C is a smooth curve that begin at the point (2, 1 ) and ends at (0,3). (cos t, sin t) fromtto t particle that moves along the curve. (Write the value of work done without evaluating (d) Find the work done by the force field F(r,...
A particle moves on a counter clockwise, origin-centered circular path in the x y-plane at a...
A particle moves on a counter clockwise, origin-centered circular path in the x y-plane at a constant rate. The radius of the circle is r, the mass of the particle is m, and the particle completes one revolu tion in time τ a) Neatly draw the following things 1. The path of the particle 2. A dot on the path when the particle is at 00 90° and 210° where 0 is mea sured from the x-axis (posi tive counter-clockwise)...
Let a, b and c be constants and let the force field be given by F(x,y,z)...
Let a, b and c be constants and let the force field be given by F(x,y,z) = ax i+by j+cz k. If the work done by the force field F on a particle as it moves along a curve given by r(t) = costi +te'sint j+tk 312 .Osts it, is equal to . Find the value of the constant c. 4 Answer:
A force F⃗ of magnitude F making an angle θ with the x axis is applied...
A force F⃗ of magnitude F making an angle θ with the x axis is
applied to a particle located along axis of rotation A, at
Cartesian coordinates (0,0) in the figure. The vector F⃗ lies in
the xy plane, and the four axes of rotation A, B, C, and D all lie
perpendicular to the xy plane.
A particle is located at a vector position r⃗ r→r_vec with
respect to an axis of rotation (thus r⃗ r→r_vec points from...
4. Let F(x,y) - PiQj be a smooth plane vector field defined for (x,y) f (0,0), and F - dr for int...
4. Let F(x,y) - PiQj be a smooth plane vector field defined for (x,y) f (0,0), and F - dr for integer j, and all suppose Q - Py for (z, y) (0,0). In the following L-JF dr for integer j, and all G are positively oriented circles. Suppose h = π where G is the circle x2 + y2-1. (a) Find 12 for G : (x-2)2 + y-1. Explain briefly. (b) Find Is for Cs: ( -2)y 9. Explain...
(1) Let G(,y, z) = (x,y, z). Show that there exists no vector field A : R3 -> R3 such that curl(A) Hint: compute its...
(1) Let G(,y, z) = (x,y, z). Show that there exists no vector field A : R3 -> R3 such that curl(A) Hint: compute its divergence G. (2) Let H R3 -> R3 be given as H(x,y, z) = (1,2,3). Find a vector potential A : R3 -> R3 such that curl(A) smooth function = H. Show that if A is a vector potential for H, then so is A+ Vf, for any f : R5 -> R (3) Let...
A force acting on a particle in the xy plane at coordinates (x, y) is given by vector F = (F_0/r) (y hat i - x hat j), where F_0 is a positive constant and r is the distance of the particle from the origin. Show that the magnitude of this force is F_0. Show that the direction of vector F is perpendicular to vector r = x hat i + y hat j.
9. (8 marks) Let F be an inverse force field given by k F(r,y,z) r13 r, where k is a constant. Find the work done by F on an object as its point of application moves along the -axis from A(1,0, 0) to B(2,0,0)
9. (8 marks) Let F be an inverse force field given by k F(r,y,z) r13 r, where k is a constant. Find the work done by F on an object as its point of application moves...
A particle is constrained to move in the x - y plane. Its position as a function of time is given by x(t) = x_0 sin(w_xt) and y(t) = y_0 cos(w_yt). Find the force vector F on the particle. Under which condition is the force a central force (i.e., points in the vector r direction)? Find the potential energy for the general force that you found in part (a) (i.e., not necessarily central). Show that the particle's energy is conserved.
2. Consider a charged particle in a magnetic field. Let -e = -1.60 x 10-191C) be an electron, under B=0.1T) of a uniform magnetic field directed out of the page direction (o-direction). At t = 0[sec] the particle is at the origin, and the initial velocity is toward positive y direction. (a) Sketch the initial configuration of this electron (-e at the origin) into (x,y)-plane. Applying the right hand rule, indicate the magnetic force on this moving electron. Note that...
. Let F(a,y)-(3e +secztan,e -90 (a) Show that F is a conservative. (b) Find a function f (potential function) show that F (c) Use above result to evaluate Jc F. V. dr, where C is a smooth curve that begin at the point (2, 1 ) and ends at (0,3). (cos t, sin t) fromtto t particle that moves along the curve. (Write the value of work done without evaluating (d) Find the work done by the force field F(r,...
A particle moves on a counter clockwise, origin-centered circular path in the x y-plane at a constant rate. The radius of the circle is r, the mass of the particle is m, and the particle completes one revolu tion in time τ a) Neatly draw the following things 1. The path of the particle 2. A dot on the path when the particle is at 00 90° and 210° where 0 is mea sured from the x-axis (posi tive counter-clockwise)...
Let a, b and c be constants and let the force field be given by F(x,y,z) = ax i+by j+cz k. If the work done by the force field F on a particle as it moves along a curve given by r(t) = costi +te'sint j+tk 312 .Osts it, is equal to . Find the value of the constant c. 4 Answer:
A force F⃗ of magnitude F making an angle θ with the x axis is
applied to a particle located along axis of rotation A, at
Cartesian coordinates (0,0) in the figure. The vector F⃗ lies in
the xy plane, and the four axes of rotation A, B, C, and D all lie
perpendicular to the xy plane.
A particle is located at a vector position r⃗ r→r_vec with
respect to an axis of rotation (thus r⃗ r→r_vec points from...
4. Let F(x,y) - PiQj be a smooth plane vector field defined for (x,y) f (0,0), and F - dr for integer j, and all suppose Q - Py for (z, y) (0,0). In the following L-JF dr for integer j, and all G are positively oriented circles. Suppose h = π where G is the circle x2 + y2-1. (a) Find 12 for G : (x-2)2 + y-1. Explain briefly. (b) Find Is for Cs: ( -2)y 9. Explain...
(1) Let G(,y, z) = (x,y, z). Show that there exists no vector field A : R3 -> R3 such that curl(A) Hint: compute its divergence G. (2) Let H R3 -> R3 be given as H(x,y, z) = (1,2,3). Find a vector potential A : R3 -> R3 such that curl(A) smooth function = H. Show that if A is a vector potential for H, then so is A+ Vf, for any f : R5 -> R (3) Let...
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