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Prove the “parallel axis theorem”: The moment of inertia I of a body about a given axis is
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Mathematical Methods in the Physical Sciences
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- For the sphers x-12+y+22+z-42=36 and x2+y2+z2=64, find the ratio of their a surface areas. b volumes.arrow_forwardA soda can is made from 40 square inches of aluminum. Let x denote the radius of the top of the can, and let h denote the height, both in inches. a. Express the total surface area S of the can, using x and h. Note: The total surface area is the area of the top plus the area of the bottom plus the area of the cylinder. b. Using the fact that the total area is 40 square inches, express h in terms of x. c. Express the volume V of the can in terms of x.arrow_forwardSolve these prism and cylinder exercises. Where necessary, round the answers to 2 decimal places unless otherwise specified. A solid right cylinder 9.55 centimeters high contains 1910 cubic centimeters of material. Compute the cross-sectional area of the cylinder.arrow_forward
- The acceleration due to gravity, g, varies with height above the surface of the earth, in a certain way. If you Go down below the surface of the earth, g varies in a different way. It can be shown that g is given by GMr for rR where Ris the radius of the earth, M is the mass of the earth, G is the gravitational constant, and ris the distance to the center of the earth. a) Sketch a graph of g against r. GM R GM R2 GM R R R R Figure 1 Figure 2 Figure 3 Choose the correct graph number from the table above: b) Is ga continuous function of r? c) Is ga differentiable function of r?arrow_forwardUse the concept of Particular Antiderivatives and Rectilinear Motion to answer the problem below: On the edge of an 8m building, Sam throws his toy vertically upwards at initial velocity = 28 m/s. If the only force that acts on the toy is accelerationdue to gravity (-32 ft/s), a. How many seconds will it take for the toy to reach its maximum height?b. What is the toy's speed when it hits the ground?arrow_forwardTwo particles have a mass of 7 kg and 15 kg, respectively. Part A If they are 800 mm apart, determine the force of gravity acting between them. Express your answer to three significant figures and include the appropriate units.arrow_forward
- A certain force (x) is applied to a wire and the elongation of the wire is recorded (y). The following data were recorded: Force (x) Elongation (y) 1 24.5 2 40 46 14 81.5 70 89.5 a. Find the sum of the Xs. b. Find the sum of the Ys. C. Find the sum of the Xs squared. d. Find the sum of the Ys squared. e. Find the sum of the XYs. f. Find the slope of the line of best fit. g. Find the y intercept.arrow_forwardDefine the relationship between x and yarrow_forward"Hand-in question": While biking, the force of air resistance F (mass times length, per time- squared) appears to be positively related to the speed v (length per time) at which the bike travels and the cross-sectional area A (length-squared) of bike and biker. However, the force involves mass and hence cannot depend on v and A alone. Let the force also depend on the air density p (mass per length-cubed). Hence, the force F can be expressed as some multiplicative combination of v, A and p, that is F = kv" Aª p², for k E R. Find the values of r, y and z that result in the above equation being dimensionally consistent.arrow_forward
- What must be the value of y if u = (1/3, y, -1/2), v = (6, 1/8, 10), and u · v = -5/2. 1 4arrow_forwardIn conclusion, state the relationship between the mass of the spherical object and the distances they were let to roll with reference to the ramp to the knock-back distance of catcher 1. What implications do these relationships give concerning the magnitude of the force which causes the knocking-back of catcher 1?arrow_forward4. Your upper arm exerts 60N and lower arm exerts 40N anteriorly and superiorly. When your upper and lower arms are bent at 90°, describe the magnitude and direction of the resultant force exerted on your arm. nff a long jumper has a forward (horizontal) velocity of 15m/sec and a vertical ntall and the magnitude of thearrow_forward
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