Dynamics & Equilibrium Week 5

Contents Overview Overview Gravitational Force Gravitational Force Equilibrium Equilibrium –FBD Pulleys Pulleys Normal Force Normal Force Friction Friction Incline Incline Review Review

Overview Newton’s Laws Inertia Inertia F = ma F = ma F 1 + F 2 = 0 F 1 + F 2 = 0

Gravitational Force When objects are near the earth’s surface, the gravitational force is F = mg. When objects are near the earth’s surface, the gravitational force is F = mg. m mg

Equilibrium The system is in equilibrium. The system is in equilibrium. ∑ F = 0 T – mg = 0 T = mg m T mg

Equilibrium (cont.) ∑ F = 0 ∑ F x = 0 ∑ F y = 0 y: 2TcosӨ - mg = 0 T = mg/2cosӨ m mg TT Ө Ө

Equilibrium (cont.) m mg T2T2 T1T1 Ө1Ө1 Ө2Ө2

Free Body Diagram x: T 1 cosӨ 1 – T 2 cosӨ 2 = 0 y: T 1 sinӨ 1 + T 2 sinӨ 2 – mg = 0 Ө1Ө1 T1T1 T2T2 Ө2Ө2 mg T 1 cosӨ 1 T 1 sinӨ 1 T 2 sinӨ 2 T 2 cosӨ 2 m ∑ F = 0 ∑ Fx = 0 ∑ Fy = 0

Pulleys T is equal on both sides. T is equal on both sides. String is massless. String is massless. M > m M > m M m Mg mg T T

Pulleys FBD M: Mg – T = Ma m: T – mg = ma Mg – mg = Ma + ma (M – m)g = (M + m)a a = (M – m)g/(M + m) Mm TT Mgmg aa

Pulleys FBD (cont.) (M + m)a = Mg – mg (M + m)a = (M – m)g a = (M – m)g/(M + m) * Does not need tension! Mm a Mgmg M + m a Mgmg

Normal Force N = mg N is always orthogonal to the surface. m mg N

Friction Friction is caused by the forming and breaking of bonds between two surfaces. Friction is caused by the forming and breaking of bonds between two surfaces. It is mainly dependent on the chemistry of the materials. It is mainly dependent on the chemistry of the materials.

Friction FBD f = μN *μ is a constant, depending on the chemistry of the materials *In general, f ≤ μN. m mg N F f

Quiz 1 Apply a force F Apply a force F System in equilibrium System in equilibrium f = ? 2kg 3N

Quiz 1 f = mg f = 2(9.8) f = 20 N 2kg 3N f mg normal

Incline m Ө

Incline FBD m Ө mg N f m N f Ө mgcosӨ mgsinӨ

Incline FBD (cont.) f m N mg Ө mgcosӨ mgsinӨ System in equilibrium System in equilibriumy: f = mgsinӨ μN = mgsinӨ μ = mgsinӨ/N x: N = mgcosӨ μ = mgsinӨ/mgcosӨ μ = sinӨ/cosӨ μ = tanӨ

Review Grav. force = mg Grav. force = mg f = μN f = μN N is orthogonal to the surface. N is orthogonal to the surface. μ = tanӨ for an incline μ = tanӨ for an incline

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