SPSP 3 30 Sept 00 Name Test Group # Remember: Show all your work for full credit. Minimum of 4 steps: Draw a diagram!! What equation are you plugging into? What numbers are you substituting? What is your final answer? Ask if anything seems unclear. Vectors have magnitude and direction! or two components! Formulae and Constants: Note: bold means vector! g = 9.80 m/s = 980 cm/s G = 6.67 x 0 - N m /kg Text Eqn - vx = vx + ax t t 0 0-5 x = x + vx ( t t )+ ax t t -6 v = v + a x x 0 0 0 0 x x0 x -7 x = x0 + ( vx0 + vx) ( t t0) -8 x = x + v ( t t ) a t t 0 x 0 x 0 0 dx n dt =nxn± x n dx = xn + n + for n - f k = µ k F N f s µ s F N a c = a radial = v / r F = GM m / r W = F dr =±U F x = - du/dx K = / mv U spr = / kx p = mv F = dp/dt Impulse = F dt = p τ = r x F = r F sin θ τ = r F = r F =r p I =Σmr = r dm I = I cm + M h Page. /5. /30 3. /5 4. /35 /5
. (a) Fill in the table below by saying in words the name of the angular that corresponds to the given linear, then write the symbol for the angular, and give its units. I have filled in the first row as an example. Linear Name of angular Symbol for angular Units of angular x angular position θ radians mass Force p = momentum (b) For each of the linear equations given to the left below, write the analogous rotational equation in the column to the right below. linear equation rotational equation v x = v 0x + a x (x - x 0 ) K = / mv ΣF = dp dt ΣF = m a (c) Under what general circumstances is angular momentum conserved?.(a) The three objects shown below are all constructed from identical rectangular rods and have the same mass and the same maximum horizontal dimension. The axes of rotation are shown by the the heavy black lines on the left. Put these objects in order from the one with the largest moment of inertia to smallest: A B C Largest Smallest continues over Page
(b) Consider a thin rod of mass M and length L which can rotate around an axis located L/4 from one end. Set up the integral necessary to find the moment of inertia of the rod about the given axis. Make all of your steps clear, but it is not necessary to evaluate the integral. (c) Consider a circular hoop of mass M and radius R and negligible thickness. What is the moment of inertia of the hoop about its center of mass? (c) If the hoop is now hung off a very-thin rod as shown and allowed to rotate about a point on its edge, what is the moment of inertia of the off center hoop? rod hoop 3. A hemispherical (radius R) bowl has a rough surface with friction on its left half and a smooth frictionless surface on the right half. A nickel released from rest at height H rolls without slipping down the left half of the bowl. H (a) Write an equation which you could solve for the speed of the nickel at the lowest position. You may include the mass m and radius r of the nickel, the initial height of the nickel H and any known constants in your equation. Note: I cm = ( / )mr for the nickel. Answer may include only: m, r, H, R, known consts, v (the unknown!) (b) Does the nickel continue to rotate at it moves up the frictionless right-hand side of the bowl? yes no (Circle one.) Explain. continues over Page
4. A disk rotates about its central axis, and it accelerates with constant angular acceleration. At one time it is rotating at 37 rev/s. Then 83 revolutions later, its angular speed is 7 rev/s. (a) Calculate the angular acceleration. (b) Calculate the time required to complete the 83 revolutions mentioned. 5. A thin stainless steel disk (with moment of inertia 0.0400 kg m ) is rotating (without friction) in a horizontal plane about its center with an angular speed of.0 rad/sec. A lead block of mass 0.50 kg is dropped from a height of 50.0 cm onto the rotating disk and sticks in place (a negligible amount of instant super glue?) a distance of 5.0 cm from the axis of rotation. (a) Is mechanical energy conserved in this process? yes no (Circle one.) Why or why not? (b) Calculate the new angular speed of the disk-cube combination. continues over Page 3
6. A modified Atwood s machine consists of the following: two masses, connected as shown in the diagram, are released from rest; M is on a frictionless surface, and the cord does not slip on the pulley. The m (= 40 grams) falls with an acceleraton of 0.444 m/s. M = 350 gr and the radius of the pulley is.54 cm. (a) What is the magnitude of the acceleration of mass? Explain. 34.0 m m (b) Draw the force diagrams (include the acceleration as a double arrow ); write the appropriate Newton s nd law equations for each object; calculate the value of the moment of inertia of the pulley. Bonus: What is the approximate direction of the force supporting the pulley? m m (c) Why isn t the tension in the cord between the pulley and mass equal to the tension in the cord between the pulley and mass? E N D Page 4