This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
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a) A diagram illustrating the forces acting on the box would show: • A horizontal surface with a box on it. • The weight of the box, W = mg, acting vertically downwards from the center of the box. • The normal force, N, acting vertically upwards from the surface, perpendicular to the floor. • The applied force, F = 200 \, N, acting at an angle of 30^ below the horizontal, pushing the box. This force can be resolved into two components: • A horizontal component, F_x = F 30^, acting in the direction of motion. • A vertical component, F_y = F 30^, acting vertically downwards. • The kinetic friction force, f_k = N, acting horizontally opposite to the direction of motion. b) Step 1: Identify the conditions for constant speed. Since the box moves at a constant speed, its acceleration is zero. This means the net force acting on the box in both the horizontal and vertical directions is zero. F_x = 0 F_y = 0 Step 2: Resolve forces in the vertical direction. The forces acting vertically are the normal force (N) upwards, the weight (mg) downwards, and the vertical component of the applied force (F 30^) downwards. N - mg - F 30^ = 0 N = mg + F 30^ Step 3: Resolve forces in the horizontal direction. The forces acting horizontally are the horizontal component of the applied force (F 30^) in the direction of motion and the kinetic friction force (f_k = N) opposite to the direction of motion. F 30^ - f_k = 0 F 30^ - N = 0 Step 4: Substitute the expression for N into the horizontal force equation and solve for m. We are given F = 200 \, N, = 0.4, = 30^, and g = 10 \, m/s^2. F 30^ - (mg + F 30^) = 0 Substitute the known values: 200 30^ - 0.4 (m · 10 + 200 30^) = 0 Using 30^ = sqrt(3)2 and 30^ = (1)/(2): 200 (sqrt(3)2) - 0.4 (10m + 200 · (1)/(2)) = 0 100sqrt(3) - 0.4 (10m + 100) = 0 100sqrt(3) - 4m - 40 = 0 Rearrange to solve for m: 4m = 100sqrt(3) - 40 m = 100sqrt(3) - 404 m = 25sqrt(3) - 10 Calculate the numerical value: m ≈ 25(1.732) - 10 m ≈ 43.3 - 10 m ≈ 33.3 \, kg The value of m is 33.3 \, kg. That's 3 down. 2 left today — send the next one.