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4. a. A buzzing fly moves in a helical path given by the equation Show that...
4. A bee goes out from its hive in a spiral path given in plane polar...
4. A bee goes out from its hive in a spiral path given in plane polar coordinates by r= bekt A = ct where b, c and k are positive constants. Show that the angle between the velocity and acceleration vectors remains constant as the bee moves outward (Hint: Find v.a/va).
2. A particle moves in the x-y plane. Its coordinates are given as functions of time...
2. A particle moves in the x-y plane. Its coordinates are given as functions of time t(2 0) b x(t)-R(at-sina)t), )Sketch the trajectory of the particle. This is the trajectory of a point on the rim of a wheel y(t)-R(1-cosω t), where R and ω are constants. (a) (3 that is rolling at a constant speed on a horizontal surface. The curve traced out by such a point as it moves through space is called a cycloid. (b) (5 Find...
CYLINDRICAL COMPONENTS OF VELOCITY: The airplane on the amusement park ride moves along a path...
The airplane on the amusement park ride moves along a path defined by the equations r = 4 m, θ = (0.2t)rad, and z = (0.5cosθ)m, where t is in seconds.A)Determine the cylindrical components of the velocity of the airplane when t = 6s.B)Determine the cylindrical components of the acceleration of the airplane when t = 6s.
You are given the metric of a 4-dimensional Einstein universe in terms of spherical polar coordinates r, 0, ø, -dt2e20(...
You are given the metric of a 4-dimensional Einstein universe in terms of spherical polar coordinates r, 0, ø, -dt2e20(1) dr2 +r2 (d02 + sin2 0dp2), ds2 (a) Use the variational principle 2 ds dA 0, dA to find the geodesic equation for r with parameter A, between given points X1 and A2. (b) Deduce or otherwise show that Г — Ф", Ге — —re 20, To= -rsin2 Oe-24 rr =0 if uv and T T
You are given the...
Prob. 3 If a 2-kg particle moves along a path given by: r(t) = 5 m...
Prob. 3 If a 2-kg particle moves along a path given by: r(t) = 5 m = constant, and (35) e(t) = 2t rads with t in seconds (36) Sketch and name the entire motion path, and determine the magnitude of the net force and acceleration at any time that you choose.
Problem H1.B Given: A particle P travels on a path described by the Cartesian coordinates of...
Problem H1.B Given: A particle P travels on a path described by the Cartesian coordinates of y ca(b - ) where and y have the units of meters.The -component of velocity, i, for P is constamt. Find: For this problem (a) Make a sketch of the path of P over the range of 0 <b. (b) Determine the Cartesian components of the velocity and acceleration of P at 0. Add sketch of the velocity and acceleration vectors for P to...
A particle moves so that its position vector is given by r -coswt i+sin wt j...
A particle moves so that its position vector is given by r -coswt i+sin wt j where w is constant a. Show that the velocityof the particle is perpendicular tor . b. Find the magnitude of acceleration in the direction of 2-h C. Show that dt
An object moves in a circular path with constant angular speed. Compare the direction of the...
An object moves in a circular path with constant angular speed. Compare the direction of the object's tangential velocity vector to its centripetal acceleration vector. A) Both vectors point in the same direction B) The vectors point in opposite directions C) The vectors are perpendicular D) Impossible. The acceleration is zero. Two objects with masses m1 and m2 are original a distance r apart. The magnitude of the gravitational force between them is F. if BOTH masses are doubled, and...
1. A particle of unit mass moves along a trajectory , 2r) θ E (03), and θ E ( a coal, -a cose r(8...
Acceleration in polar coordinates is required
1. A particle of unit mass moves along a trajectory , 2r) θ E (03), and θ E ( a coal, -a cose r(8)--, expressed in plane polar coordinates. The angle 6(t) changes with time according to the equation θ wt. Here a, are positive constants independent of time. (a) [10 marks) Compute the transverse acceleration of the particle (b) [10 marks) Find the force acting on a particle and express it in terms...
One important quantity to characterise a fluid is the circulation Given Γ, a closed oriented path...
One important quantity to characterise a fluid is the circulation Given Γ, a closed oriented path inside the fluid, the circulation C, along the path Γ is the integral a) Consider a cylindrical reservoir of radius r containing a fluid rotating with constant angular velocity ω. What is the circulation Cr with path defined as in figure ? Figure 1 b) Now, let us consider the case in which the velocity is not constant but rather the circulation, around any...
4. A bee goes out from its hive in a spiral path given in plane polar coordinates by r= bekt A = ct where b, c and k are positive constants. Show that the angle between the velocity and acceleration vectors remains constant as the bee moves outward (Hint: Find v.a/va).
2. A particle moves in the x-y plane. Its coordinates are given as functions of time t(2 0) b x(t)-R(at-sina)t), )Sketch the trajectory of the particle. This is the trajectory of a point on the rim of a wheel y(t)-R(1-cosω t), where R and ω are constants. (a) (3 that is rolling at a constant speed on a horizontal surface. The curve traced out by such a point as it moves through space is called a cycloid. (b) (5 Find...
You are given the metric of a 4-dimensional Einstein universe in terms of spherical polar coordinates r, 0, ø, -dt2e20(1) dr2 +r2 (d02 + sin2 0dp2), ds2 (a) Use the variational principle 2 ds dA 0, dA to find the geodesic equation for r with parameter A, between given points X1 and A2. (b) Deduce or otherwise show that Г — Ф", Ге — —re 20, To= -rsin2 Oe-24 rr =0 if uv and T T
You are given the...
Prob. 3 If a 2-kg particle moves along a path given by: r(t) = 5 m = constant, and (35) e(t) = 2t rads with t in seconds (36) Sketch and name the entire motion path, and determine the magnitude of the net force and acceleration at any time that you choose.
Problem H1.B Given: A particle P travels on a path described by the Cartesian coordinates of y ca(b - ) where and y have the units of meters.The -component of velocity, i, for P is constamt. Find: For this problem (a) Make a sketch of the path of P over the range of 0 <b. (b) Determine the Cartesian components of the velocity and acceleration of P at 0. Add sketch of the velocity and acceleration vectors for P to...
A particle moves so that its position vector is given by r -coswt i+sin wt j where w is constant a. Show that the velocityof the particle is perpendicular tor . b. Find the magnitude of acceleration in the direction of 2-h C. Show that dt
Acceleration in polar coordinates is required
1. A particle of unit mass moves along a trajectory , 2r) θ E (03), and θ E ( a coal, -a cose r(8)--, expressed in plane polar coordinates. The angle 6(t) changes with time according to the equation θ wt. Here a, are positive constants independent of time. (a) [10 marks) Compute the transverse acceleration of the particle (b) [10 marks) Find the force acting on a particle and express it in terms...
One important quantity to characterise a fluid is the circulation Given Γ, a closed oriented path inside the fluid, the circulation C, along the path Γ is the integral a) Consider a cylindrical reservoir of radius r containing a fluid rotating with constant angular velocity ω. What is the circulation Cr with path defined as in figure ? Figure 1 b) Now, let us consider the case in which the velocity is not constant but rather the circulation, around any...
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