Various Math #66 - Q&A

Question 1 When the wagon is at rest 50 meters up a 6.0 degree slope, an outlaw cuts it loose from the hoses with two boxes of gold weighing a combined 300 kg. The bandit intends to cause the wagon to crash into the canyon where his comrades are waiting after rolling down the slope and across the level terrain. But the Lone Ranger (mass 75.0kg) and Tonto are waiting in a tree 40 meters from the canyon's brink (mass60.0kg). They watch as it passes them below and then dive vertically into the wagon. Will they make it before the wagon overturns if it takes them 5.0 seconds to grab the gold and get out? b) Is the kinetic energy of the system comprising the two heroes plus the wagon preserved as they fall into the wagon? If not, does it change and if so, how much? Answer h = 50sin (6.0) = 5.23m mgh = 1/2 mv² v = 10.1m/s m₀v₀ = mf vf Vi/vf = mc/mf vf = m₀ v₀ / mf vf = 6.97/5 = 34.8 m/s b) 1/2 m₀ v₀² = 1/2 * mf vf² = 1/2(mi *vᵢ² - mf vf²)= 4740 J Kinetic energy dropped. Question 2 (a) A 115-g mass of aluminum is formed into a right circular cylinder, shaped so that its diameter equals its height.Calculate the resistance between the top and bottom faces of thecylinder at 20 degrees C. (b) Calculate the resistance betweenopposite faces if the same mass of aluminum is formed into acube. Answer Given the mass of the aluminium is m = 115 10⁻³ kg Density of the aluminium is ρ = 2.70 * 10³ kg/m ³ The specidic resistance of the aluminium is ρ = 2.8210⁻⁸ Ohm/m Now the volume of the block is volumeV = m/ρ = (115 × 10⁻³) / (2.70 × 10³) = 4.2610⁻⁵ m³ a) Given the diameter is equal to its height d = h We know the formula for the volume of the cylinder in terms of diameter is V = πr²h = π(d/2)²h = π(d/2)²d = πd³/4 From this d = (4V/π)⅓ = (4(4.26 × 10⁻⁵) / π)⅓ = 0.03785m R= ρl/A = ρd/ π(d/2)² = 4ρ/πd We know the formula for the resistance is R = 4(2.82 × 10⁻⁸) / π(0.03785) = 9.49 x 10⁻⁷ Ohms b) In this case we have to calculate the resistance between theopposite faces of the aluminium cube. We know the formula for the cube is V = L³ From this the length of the cube is L = ∛V = ∛(4.26 × 10⁻⁵) = 0.0349 Now the resistance between the opposite faces of the cubeis R = ρL/A = ρL / L² = ρ/L = 2.82 × 10⁻⁸ / 0.0349 = 8.07 x 10⁻⁷ Ohms Question 3 A picture hangs on a nail as shown below. The tension T ineach string segment is 3.5 Ν. A) What is the equilibrant or the upward reaction force of the nail B) What is the wieght of the picture. T | T | / | 45° / | -------- Answer 1 Consider the free body diagram of the picture and resolve thetension forces alongthe horizontal and vertical directions Tcos45 | 2Tsin45| | Tcos45 -------- Use code with caution. The value of 2Tsin 45 is balanced by the weight of the picture. 2Tsin 45 = mg and the total reaction forcre on the nail is 2Tsin 45 Answer 2 The nail is stationary, hence the total force on the nail is zero. The picture is hanging on the nail and the whole system is stationary so the net force is zero. The formula to calculate the normal force on the nail from the figure is N = T sin 45° + T sin 45° where N is the normal force of the nail and T is the tension in the string. Substitute 3.5 N for T into the formula to calculate the normal force on the nail N = 3.5N x sin 45° + 3.5N x sin 45° = 4.95 N Question 4 A 10 kg objectexperiences a horizontal force which causes it to accelerate at 5 m/s², moving it a distance of 20 m, horizontally.How much work is done by the force? A ball is connected to a rope and swung around in uniform circular motion. The tension in the rope is measured at 10 N and the radius of thecircle is 1 m. How much work is done in one revolution around the circle? A 10 kg weight issuspended in the air by a strong cable. How much work is done, perunit time, in suspending the weight? A 5 kg block is moved up a 30 degree incline by a force of 50 N, parallel to the incline. The coefficient of kinetic friction between the block and the incline is .25. How much work is done by the 50 N force in moving the block a distance of 10 meters? What is the total workdone on the block over the same distance? What is the kinetic energy of a 2 kg ball that travels a distance of 50 metersin 5 seconds? A ball is thrown vertically with a velocity of 25 m/s. How high does it go? What is its velocity when it reaches a height of 25 m? A ball with enough speed can complete a vertical loop. With what speed must the ballenter the loop to complete a 2 m loop? (Keep in mind that the velocity of the ball is not constant throughout the loop). Answer A horizontal force causes an item weighing 10 kg to accelerate at 5 m/s², moving it 20 meters in a horizontal direction. How much labor does the force perform? The force's magnitude is determined by F=ma = (10) (5) = 50N It acts over a 20-meter distance in the same direction as the object's displacement, indicating that the force's total work is provided by W=F*x = (50) (20) = 1000 Joules Question 5 C Common salt, NaCl, has a density of 2.165 g/cm³. The molecular weight of NaCl is 58.44. Estimate the distance between nearest neighbor Na and Cl ions. (Hint: each ion can be considered to have one cube or cell of side s (our unknown) extending out from it) Answer The distance between nearest neighbouring ion in NaCl d³ = M / 2Nρ d = [M / 2Nρ]⅓ M - Molecular weight N - avagadro number P - density d = 2.823A°