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Quiz * Vector A has a magnitude of 4 units and makes an angle of 37° with the positive x-axis. Vector B has a magnitude of 6 units and makes an angle of 120° with the positive x- axis. a)Find the x-component and the y-component of each vector. b)Find A – B.

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Two vectors are given by A = 3i + 3j, and B = -2i + 7j. Find 1. A + B, 2. A – B, 3. Find the angle between the two vectors

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3.2.1The first condition for Equilibrium: For an object to be at rest, Newton's second law tells us that the sum of the forces acting on it must add up to zero. Since force is a vector, the components of the net force must each be zero. Hence, a condition for equilibrium is that ∑ Fx = 0, ∑Fy = 0, ∑Fz = 0 3.4 ∑ Fx = 0, ∑Fy = 0, ∑Fz = 0 3.4 Equation 3.4 are called the First Condition for Equilibrium.

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Example 3.2: For chandelier cord tension. Calculate the tensions FA and FB in the two cords that are connected to the vertical cord supporting the 200kg chandelier in Fig 3.2. Solution: We first resolve FA into its horizontal (x) and vertical (y) components. Although we don’t know the value of FA. We can write, FAx = - FAcos60° and FAy=FAsin60° FB has only an x – component. In the vertical direction, we have the downward force exerted by the vertical cord equal to the weight of the chandelier, and the vertical component of FA upward. The weight of the chandelier is W = mg = 200 × 9.8 = 1960N = 200 × 9.8 = 1960N Since, ∑Fy = 0, we have ∑Fy =, FA sin 60 – w = 0 FA sin 60 – 1960 = 0 FA sin 60 – 1960 = 0

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So, In the horizontal direction, ∑ Fx = FB – FA cos 60° = 0 FB = FA cos 60 = 2263 cos60 = 1131 N The magnitude of FA and FB determine the strength of cord or wire that must be used.

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3.2.2 The Second Condition For Equilibrium The second condition for equilibrium: that the sum of the torques acting on an object, as calculated about any axis, must be zero: ∑τ = 0 (3.5) ∑τ = 0 (3.5)

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Example 3.3: For Balancing a seesaw. A board of mass M=2.0kg serves as a seesaw for two children, as shown in Fig3.3a, child A has a mass of 30kg and sits 2.5m from the pivot point, P (his center of gravity is 2.5m from the pivot). At what distance x from the pivot must child B, of mass 25kg, place herself to balance the seesaw? Assume the board is uniform and centered over the pivot. Solution: We draw free body diagram, as shown Fig.3.5bk The First: force equation. All the forces are in the y (vertical) direction, so ∑Fy = 0, FN – mAg – mBg – Mg = 0 Where FA = mAg and FB = mBg, because each child is in equilibrium when the seesaw is balanced. The Second: Torque equation is ∑τ = 0, mAg (2.5m) – mBg + Mg(0m) + FN (0m) = 0 or, mAg (2.5m) – mBgx = 0 then To balance the seesaw, child B must sit so that her CM is 3.0m from the pivot point.

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Example 3.4: For forces an a beam and supports. A uniform 1500 kg beam, 20.0m long, supports a 15,000 kg printing press 5.0m from the right support column Fig. 3.6. Calculate the force on each of the vertical support columns. Solution: The torque, ∑τ = 0, with the counterclockwise direction as positive gives ∑τ = - (10.0m) (1500kg)g – (15.0m) (15,000kg)g + (20.0m)FB = 0 Solving for FB, we find = (12,000 kg)g = 118,000N. To find FA, we use ∑Fy = 0, with + y upward ∑Fy = FA – (1500 kg)g – (15,000 kg)g + FB = 0 Putting in FB = (12,000 kg)g, we find that FA = (4500 kg)g = 44,100 N

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Example 3.5: For Hinged beam and cable. A uniform beam, 2.20m long with mass m = 25.0 kg, is mounted by a hinge on a wall as shown in Fig 3.7. The beam is held in a horizontal position by a cable that makes an angle θ = 30.0° as shown. The beam supports a sign of mass M = 28.0 kg suspended from its end. Determine the components of the force FH that the hings exerts on the beam, and the tension FT in the supporting cable. Example 3.5: For Hinged beam and cable. A uniform beam, 2.20m long with mass m = 25.0 kg, is mounted by a hinge on a wall as shown in Fig 3.7. The beam is held in a horizontal position by a cable that makes an angle θ = 30.0° as shown. The beam supports a sign of mass M = 28.0 kg suspended from its end. Determine the components of the force FH that the hings exerts on the beam, and the tension FT in the supporting cable. Solution: The sum of the forces in the vertical (y) direction is ∑Fy = 0, ∑Fy = 0, FHy + FTy – mg – Mg = 0 (i) In the horizontal (x) direction, the sum of the forces is ∑Fx = 0, ∑Fx = 0, FHx - FTx = 0 (ii) FHx - FTx = 0 (ii) For the torque equation, we choose the axis of the point where FT and Mg act. We choose torques that tend to rotate the beam counterclockwise as positive. The weight mg of the beam acts at its center, so we have

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∑τ = 0 - (FHy) (2.20 m) + mg (1.10m) = 0 We solve for FHy: Next, since the tension FT in the cable acts along the cable (θ = 30.0°). We see from Fig 3.7 that, or FTy = FTx tanθ = FTx tan30° = 0.577 FTx (iv) Equation (i) above gives; FTy= (m+M)g – FHy = (53.0kg) (9.8m/s2) - 133N = 396N Equation (iv) and (ii) give; FHx = FTx = 687N The components of FH are FHy = 123N and FHx = 687 N. the tension in the wire is

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Example 3.5: For ladder. A 5.0m – long ladder leans a against a wall at a point 4.0m above a cement floor as shown in Fig 3.7. The ladder is a uniform and has mass m= 12.0kg. Assuming the wall is frictionless (but the floor is not)k determine the forces exerted on the ladder by the floor and by the wall. We use the equilibrium conditions, ∑Fx = 0, ∑Fy = 0, ∑τ = 0 We will need all three since there are three unknowns: FW, Fcx, and Fcy. The y component of the force equation is The y component of the force equation is ∑Fy = Fcy - mg = 0 So immediately we have Fcy = mg = 118N The x component of the force equation is ∑Fx = Fcx - FW = 0 To determine both Fcx and FW, we need a torque equation. If we choose to calculate torques about an axis through the point where the ladder touches the cement floor, then Fc, which acts at this point, will have a lever arm of zero and so won’t enter the equation. The ladder touches the floor a distance;

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The lever arm for mg is half this, or 1.5m, and the lever arm for FW is 4.0m, Fig3.7, we get ∑τ = (4.0m)FW – (1.5m)mg = 0 Thus, Then, from the x – component of the force equation, Fcx = FW = 44N Since the components of Fc are Fcx = 44N and Fcy = 118N, then And it acts at an angle to the floor of

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