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Angular Mechanics - Rolling using dynamics Contents: Review Linear and angular Qtys

Angular Mechanics - Rolling using dynamics Contents: Review Linear and angular Qtys Tangential Relationships Useful Substitutions Force causing  Rolling | Whiteboard Strings and pulleys Example | Whiteboard. Angular Mechanics - Angular Quantities. Linear: (m) s (m/s) u

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Angular Mechanics - Rolling using dynamics Contents: Review Linear and angular Qtys

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  1. Angular Mechanics - Rolling using dynamics • Contents: • Review • Linear and angular Qtys • Tangential Relationships • Useful Substitutions • Force causing  • Rolling | Whiteboard • Strings and pulleys • Example | Whiteboard

  2. Angular Mechanics - Angular Quantities Linear: (m) s (m/s) u (m/s) v (m/s/s) a (s) t (N) F (kg) m Angular:  - Angle (Radians) o - Initial angular velocity (Rad/s)  - Final angular velocity (Rad/s)  - Angular acceleration (Rad/s/s) t - Uh, time (s)  - Torque I - Moment of inertia TOC

  3. Angular Mechanics - Tangential Relationships Linear: (m) s (m/s) v (m/s/s) a Tangential: (at the edge of the wheel) = r - Displacement = r - Velocity = r - Acceleration* *Not in data packet TOC

  4. Angular Mechanics - Useful Substitutions  = I  = rF so F = /r = I/r s = r, so  = s/r v = r, so  = v/r a = r, so  = a/r TOC

  5. Angular Mechanics - Force causing  F r  = I  = rF so F = /r = I/r TOC

  6. Angular Mechanics - Rolling mgsin I = 1/2mr2 m r - cylinder  F = ma + /r mgsin = ma + I/r ( = I) mgsin = ma + (1/2mr2)(a/r)/r ( =a/r) mgsin = ma + 1/2ma = 3/2ma gsin = 3/2a a = 2/3gsin TOC

  7. Whiteboards: Rolling 1 | 2 | 3 TOC

  8. A marble (a solid sphere) has a mass of 23.5 g, a radius of 1.2 cm, and rolls 2.75 m down a 21o incline. Solve for a in terms of g and  mgsin = ma + I/r, I = 2/5mr2,  = a/r mgsin = ma + (2/5mr2)(a/r)/r mgsin = ma + 2/5ma = 7/5ma gsin = 7/5a a = 5/7gsin W 5/7gsin

  9. A marble (a solid sphere) has a mass of 23.5 g, a radius of 1.2 cm, and rolls 2.75 m down a 21o incline. Plug in and get the actual acceleration. (a = 5/7gsin) a = 5/7gsin = 5/7(9.8m/s2)sin(21o) a = 2.5086 m/s2 = 2.5 m/s2 W 2.5 m/s/s

  10. A marble (a solid sphere) has a mass of 23.5 g, a radius of 1.2 cm, and rolls 2.75 m down a 21o incline. What is its velocity at the bottom of the plane if it started at rest? (a = 2.5086 m/s2) v2 = u2 + 2as v2 = 02 + 2(2.5086 m/s2)(2.75 m) v = 3.714 m/s = 3.7 m/s W 3.7 m/s

  11. Angular Mechanics – Pulleys and such r m1 m2 • For the cylinder: •  = I • rT = (1/2m1r2)(a/r) • (Where T is the tension in the string) • For the mass: • F = ma • m2g - T = m2a TOC

  12. Angular Mechanics – Pulleys and such r m1 m2 • So now we have two equations: • rT = (1/2m1r2)(a/r) or • T = 1/2m1a • and • m2g - T = m2a TOC

  13. Angular Mechanics – Pulleys and such r m1 m2 • T = 1/2m1a • m2g - T = m2a • Substituting: • m2g - 1/2m1a = m2a • Solving for a: • m2g = 1/2m1a + m2a • m2g = (1/2m1 + m2)a • m2g/(1/2m1 + m2) = a TOC

  14. Whiteboards: Pulleys 1 | 2 | 3 | 4 TOC

  15. r m1 m2 A string is wrapped around a 12 cm radius 4.52 kg cylinder. A mass of .162 kg is hanging from the end of the string. Set up the dynamics equation for the hanging mass. (m2) m2g - T = m2a W figure it out for yourself

  16. r m1 m2 A string is wrapped around a 12 cm radius 4.52 kg thin ring. A mass of .162 kg is hanging from the end of the string. Set up the dynamics equation for the thin ring (m1)  = I, I = m1r2,  = a/r,  = rT rT = (m1r2)(a/r) T = m1a W figure it out for yourself

  17. r m1 m2 A string is wrapped around a 12 cm radius 4.52 kg thin ring. A mass of .162 kg is hanging from the end of the string. Solve these equations for a: m2g - T = m2a T = m1a m2g - m1a = m2a m2g = m1a + m2a m2g = a(m1 + m2) a = m2g/(m1 + m2) W figure it out for yourself

  18. r m1 m2 A string is wrapped around a 12 cm radius 4.52 kg thin ring. A mass of .162 kg is hanging from the end of the string. Plug the values in to get the acceleration: a = m2g/(m1 + m2) a = m2g/(m1 + m2) a = (.162 kg)(9.80 N/kg)/(4.52 kg + .162 kg) a = .339 m/s/s W .339 m/s/s

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