Forces Connected Particles KUS objectives BAT Set up

Forces: Connected Particles • KUS objectives BAT Set up a model of two connected particles with uniform tension on an inelastic string/cable Starter: What will be true about the Tension in a string/rope that joins two objects? Moo Tension is equal at both ends

WB 1 The diagram shows a cow hanging by means of a cable and pulley and held in equilibrium by a weight with mass 250 kg, the mass of the cow is 200 kg. Draw a force diagram and find all the forces in a model of this situation n o i t c ea R n o i ns Te n o i t c i r F 200 Weight (W) Tension

WB 1 connected particles in Equilibrium For the COW So Tension T = 1960 N For the Weight n o i t ac Re on i t c ri F Weight = 250 g = 2450 N n sio n e T Tension = 1960 N 200 2450 cos 20 = 2302. 2 Friction = 1960 – 837. 9 Friction = 1122. 1 N 200 2450 sin 20 = 837. 9 Reaction = 2302. 2 N

WB 2. connected particles in Equilibrium What Forces are in operation? Tension 420 Tension Weight Tension 0 42 Weight Tension Weight Each cow has mass = 200 kg, Mass of forklift = m

WB 2 Connected Particles Find the mass of the forklift (ii) For both COWs Weight = 200 g = 1960 N So Tension = 1960 N For the forklift 420 Both Tension = 1960 N Both j components are = 1960 sin 42 = 1311. 5 N So Weight = 2623 N So mass m = 2623 9. 8 = 267. 7 kg

Newton's Three laws: A particle will remain at rest or will continue to move with constant velocity in a straight line unless acted on by a resultant force Every Action has an equal and opposite reaction The Force applied to a particle is proportional to the mass of the article and the acceleration produced F = ma

WB 3 Connected particles moving in different directions Problems concerning connected particles moving in different directions must be solved by considering the particles separately two particles are connected by a light, inextensible string over a smooth pulley Find the acceleration of the particles For P: For Q: Find the tension in the string For P, using (1): Find the force exerted by the string on the pulley 2 m P 3 m Q

WB 4 Connected particles moving in same directions Two particles P and Q of masses 6 kg and 3 kg are connected by a light inextensible string. Particle P rests on a rough horizontal table. The string passes over a smooth pulley fixed at the edge of the table and Q hangs vertically. The system starts from rest. If the coefficient of friction μ = 1/3 , find a) The acceleration of Q b) The Tension in the string c) The force exerted on the pulley Fmax: For P: For Q: Fmax P T pulley T Q W Sub in (1)

WB 4 Connected particles moving in same directions c) The force exerted on the pulley Resolving these forces using Pythagoras gives P T = 26. 1 pulley T = 26. 1 Q

WB 5 Connected particles moving in same directions A particle P of mass 5 kg lies on a smooth inclined plane of angle θ = arcsin 3/5. Particle P is connected to a particle Q of mass 4 kg by a light inextensible string which lies along a line of greatest slope on the plane and passes over a smooth peg. The system is held at rest with Q hanging vertically 2 m above a horizontal plane. The system is now released from rest. Assuming P does not reach the peg, find, to 3 sf: a) The acceleration of Q b) How long t takes for Q to hit the horizontal plane c) the total distance that P moves up the plane. a) The acceleration of Q pulley R T Fmax: T P Q For P: WP For Q: Same angle WQ

WB 5 (cont) b) How long t takes for Q to hit the horizontal plane c) the total distance that P moves up the plane. R T First, find v after 1. 92 seconds Second, find a given that there is no longer any pull from Q (Tension) P WP

WB 6 A particle A of mass 0. 8 kg rests on a horizontal table and is attached to one end of a light inextensible string. The string passes over a small smooth pulley P fixed at the edge of the table. The other end of the string is attached to a particle B of mass 1. 2 kg which hangs freely below the pulley, as shown in the diagram above. The system is released from rest with the string taut and with B at a height of 0. 6 m above the ground. In the subsequent motion A does not reach P before B reaches the ground. In an initial model of the situation, the table is assumed to be smooth. Using this model, find (a) the tension in the string before B reaches the ground, For A: For B: Sub in (1)

WB 6 (cont) (b) the time taken by B to reach the ground.

WB 7 A particle A of mass 4 kg moves on the inclined face of a smooth wedge. This face is inclined at 30° to the horizontal. The wedge is fixed on horizontal ground. Particle A is connected to a particle B, of mass 3 kg, by a light inextensible string. The string passes over a small light smooth pulley which is fixed at the top of the plane. The section of the string from A to the pulley lies in a line of greatest slope of the wedge. The particle B hangs freely below the pulley, as shown in the diagram above. The system is released from rest with the string taut. For the motion before A reaches the pulley and before B hits the ground, find (a) the tension in the string, For A: For B: Sub in (1)

(b) the magnitude of the resultant force exerted by the string on the pulley. (c) The string in this question is described as being ‘light’. (i) Write down what you understand by this description. (ii) State how you have used the fact that the string is light in your answer to part (a). (i) The string has no weight/mass (ii) The tension in the string is constant (same at A as B)

WB 8 The diagram shows two particles A and B, of mass m kg and 0. 4 kg respectively, connected by a light inextensible string. Initially A is held at rest on a fixed smooth plane inclined at 30° to the horizontal. The string passes over a small light smooth pulley P fixed at the top of the plane. The section of the string from A to P is parallel to a line of greatest slope of the plane. The particle B hangs freely below P. The system is released from rest with the string taut and B descends with acceleration g. (a) Write down an equation of motion for B. (b) Find the tension in the string. Using (a) (c) Prove that m = For A:

(d) State where in the calculations you have used the information that P is a light smooth pulley. The tension in the string is constant (same at A as B) On release, B is at a height of one metre above the ground and AP = 1. 4 m. The particle B strikes the ground and does not rebound. (e) Calculate the speed of B as it reaches the ground. P (f) Show that A comes to rest as it reaches P. When B hits the floor, A is moving at Between then and A stopping, only gravity is acting: at P

Skills H HWK H

• KUS objectives BAT Set up a model of two connected particles with uniform tension on an inelastic string/cable self-assess One thing learned is – One thing to improve is –

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