INTRINSIC COORDINATES

Transcription

1 INTRINSIC CRDINATES

2 Question 1 (**) s = 4asin m A bead of mass m is made to slide along a smooth wire, which is fied in a vertical plane, and is bent into the shape of an inverted ccloid with intrinsic equation s = 4asin, where a is a positive constant, s is measured from the origin, and is the angle the tangent to the ccloid makes with the ais. The bead is released from a point on the ccloid where s 0. a) Show that the period of the resulting oscillations is independent of the position from which the bead was released. The bead passes through the lowest point of the path with speed u. b) Determine in terms of m, u, a and g, the magnitude of the reaction of the wire on the bead as it passes through the lowest position of the wire 4a τ =, g mu R = + mg 4a

3 Question (**) A particle moves with constant speed u on the curve with intrinsic equation s = a tan, 0 <, where a is a positive constant, s is measured from the origin, and is the angle the tangent to the curve makes with the ais. Show that the magnitude of the normal component of the acceleration of the particle, t seconds after starting from the point where = 0, is a au + u t. M456-B, proof

4 Question 3 (**) s = c tan P A particle P is moving with constant speed u on the catenar with intrinsic equation s = c tan, 0 <, where c is a positive constant, s is measured from the origin, and is the angle the tangent to the catenar makes with the ais. Find an epression for the magnitude of the acceleration of the particle and hence state the maimum magnitude of its acceleration. a cu = c + s, a ma u = c

5 Question 4 (**+) ( sec ) s = 5ln tan + B A bead of mass 0.5 kg is made to slide along a smooth wire, which is fied in a horizontal plane, and is bent into the shape of a curve with intrinsic equation ( sec ) s = 5ln tan +, 0 < where a is a positive constant, s is measured from the origin, and is the angle the tangent to the curve makes with the ais. The bead is moving along the wire with constant speed 10 1 ms. Determine the magnitude of the reaction of the wire on the bead, when =. 3 R =.5 N

6 Question 5 (***) m s = asin The figure above shows a particle of mass m, which is free to slide along a smooth surface, whose vertical cross section is the curve C with intrinsic equation where a is a positive constant. s = asin, 0 < The arclength s is measured from the origin, and the angle is the angle the tangent to C makes with the positive ais, as shown in the figure above. The particle is projected from with speed point P. 1 ag and leaves the surface at the a) Find the value of s at P. b) Determine the magnitude of the acceleration at P. M456-C, s = 1 a, acceleration = g

7 Question 6 (***) s = 4asin B A bead of mass m is made to slide along a smooth wire, which is fied in a vertical plane, and is bent into the shape of an inverted ccloid with intrinsic equation s = 4asin, 0 < where a is a positive constant, s is measured from the origin, and is the angle the tangent to the ccloid makes with the ais. The bead is projected from with tangential speed 3ag. Show that when =, 4 a) the speed of the bead is ag. b) the magnitude of the reaction of the wire on the bead is 3 mg. 4 proof

8 Question 7 (***+) A ( 1) 3 = 1 B 5 17 A rollercoaster car, of mass 00 kg, is constrained to move along a rail path with Cartesian equation ( 1) 3 =. The car comes to instantaneous rest at the point A, where = 17, and immediatel begins to freel slide downwards towards the origin, as shown in the figure above. The point B, lies on the same rail path, where = 5. The car is modelled as a particle moving along a smooth rail path without an air resistance. As the car passes through B, calculate... a)... the magnitude of the acceleration of the car as it passes through B. b)... the magnitude of the normal reaction eerted b the rails onto the car M456-A, a 16.0 ms, R 33 N

9 Question 8 (***+) s = 4asin P A bead of mass m is made to slide along a smooth wire, which is fied in a horizontal plane, and is bent into the shape of a curve with intrinsic equation s = 4asin, 0 < where a is a positive constant, s is measured from the origin, and is the angle the tangent to the curve makes with the ais. The bead is released from rest from a point on the wire where s = a. Show the magnitude of the reaction of the wire on the bead, R, is given b [ ] R = 1 mg 4cos + sec 4 tan sin. 4 proof

10 Question 9 (****) P s = acos s The figure above shows a bead of mass m, modelled as a particle P, sliding freel along a smooth wire bent into the shape of the curve with intrinsic equation s = acos, 0, where a is a positive constant, s is measured from the origin, and is the angle the tangent to the curve makes with the ais. Given that the bead was projected from the highest point on the wire with tangential speed 1 ag, determine a simplified epression for the magnitude of the normal reaction between the bead and the wire, as the bead reaches. 1 ( + 1 ) mg

11 Question 10 (****) A bead B is free to slide along a smooth wire which is bent into a circle of radius a, with centre at C. The wire is fied in a vertical plane. The bead is held at a point A, where CA is horizontal, and released from rest. When the angle ACB is θ the bead has speed v. Use intrinsic coordinates ( s, ) to find the magnitude of the acceleration of the bead when θ =. 4 a = ag 5

12 Question 11 (****) m 4k sin s = g A bead of mass m is made to slide along a smooth wire, which is fied in a vertical plane, and is bent into the shape ccloidal arch with intrinsic equation where k is a positive constant. 4k sin s =, g The arclength s is measured from the origin, and the angle is the angle the tangent to the ccloid makes with the positive ais as shown in the figure above. The bead passes through the highest point of the ccloidal arch, at time t = 0, with speed k. The particle passes through the point where θ = 1, when t = T Show that T ( ) k = ln 1 +. g proof

13 Question 1 (****) m s = asin A bead of mass m is made to slide along a smooth wire, which is fied in a vertical plane, and is bent into the shape ccloidal arch with intrinsic equation s = asin, where a is a positive constant. The arclength s is measured from the origin, and the angle is the angle the tangent to the ccloid makes with the positive ais as shown in the figure above. The bead passes through the highest point of the ccloidal arch, with speed 1 ag. When the particle has travelled a distance s, its speed is v and the normal reaction from the wire to the bead is R. a) Show, with a detailed method, that i. ii. g v = s a a +. mg R = 1 4sin cos. b) Find the distance travelled b the bead b the time R = 0. M456-D, d = 1 a [solution overleaf]

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15 Question 13 (****) A particle is constrained to move on a curve with intrinsic equation s f ( ) =. It is moving in such a wa so that it has constant tangential acceleration of magnitude a, and constant radial acceleration of magnitude a, where a is a positive constant. When t = 0, s = 0, = 0 and the particle is moving with speed u. Find the intrinsic equation of the curve on which the particle is moving. u s = e 1 a

16 Question 14 (****) ɵn ŝ P A particle P is constrained to move on a path, so that its distance travelled along that path, measured from the origin, is denoted b s. The angle the tangent to P at an position on the path makes with the ais is denoted b. The unit vector along the tangent to the path at P is denoted b ŝ and the unit vector along the normal to the path at P, directed towards the centre of curvature of the path is denoted b ɵn, as shown in the figure above. Show that the acceleration of the particle is sɺ ɺɺ s sˆ + nˆ, ρ where ρ denotes the radius of curvature. proof

17 Question 15 (****) m s = 4asin A section of thin fleible gutter tpe tubing, with a smooth groove running along its length is bend into the shape of a ccloid with intrinsic equation where a is a positive constant. s = 4asin, 0 <, The ccloidal tubing is fied in a vertical plane with its verte coinciding with a Cartesian origin, where the directions of and increasing are measured as shown the above figure. The arclength s is measured from, and the angle is the angle the tangent to the ccloidal tubing makes with the positive ais also shown in the figure above. A particle of mass m is placed in the groove of the tubing at. The particle is slightl disturbed and begins to travel down the rod, where the groove keeps the particle from falling to either side of the tubing. a) Show that while the particle is still in contact with the tubing = asin. b) Show further than the particle leaves the tubing when = a. proof [solution overleaf]

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19 Question 16 (****) 5s = 8sin B A bead B is made to slide along a smooth wire, which is fied in a vertical plane, and is bent into the shape of an inverted ccloid with intrinsic equation 5s = 8sin, 0 <, where s is measured from the origin, and is the angle the tangent to the ccloid makes with the ais. The bead is projected from with tangential speed ms. Use intrinsic coordinates to find an epression, in terms of s and g, for the speed of the bead and hence show that the bead comes to rest at a cusp. You ma not consider energ conservation in this question. [ ] v = 1 g 8 5s

20 Question 17 (****) m s = 7 tan The figure above shows a particle of mass m, which is free to slide along a smooth surface, whose vertical cross section is the curve C with intrinsic equation s = 7 tan, 0 <. The arclength s is measured from the origin, and the angle is the angle the tangent to C makes with the positive ais, as shown in the figure above. The particle is released from rest from a point A on the surface, where = 1, and 4 leaves the surface at the point B. Determine the distance AB along the curved surface. d = 7

21 Question 18 (*****) particle ( ) = ln cos The figure above shows a particle which is free to slide along a smooth surface, whose vertical cross section is the curve C with equation = ln ( cos ), 0 <. The particle is projected from with speed surface at the point P. 1 3 g tangential to C and leaves the Show that the distance P along C is 1 arcosh e3. proof

22 Question 19 (*****) A right prism is fied so that its ais is horizontal. A particle is placed on the highest point of the outer smooth surface of the prism whose cross section has equation = 1 cosh, 0. The particle is slightl disturbed and begins to move in a path along the cross section of the outer surface of the prism whose equation is given above. Determine the distance the particle travels until the instant it leaves the surface. d = 3