A concrete wall tending to fall over is to be stayed by a round iron bar fixed to the wall at one end and anchored to the ground at the other end. The bar is 10 feet long and is to make an angle of 45° with the horizontal in a plane perpendicular to the wall. It is heated to a uniform temperature of 600° F. and is then quickly secured in position so as to be just tight while hot. When the bar has cooled to 60° F. it is found that its point of attachment to the wall has moved ¼ an inch horizontally. Find the tensile stress then existing in the bar, given that the coefficient of expansion of iron is \(6.8 \times 10^{-6}\) per degree F., and that Young's Modulus of Elasticity is \(30 \times 10^6\) lbs. per sq. inch. It may be assumed that the force in the bar acts along its axis.
The square \(DEE'D'\) is supported and held rigid and loads are applied to the structure as shown in the figure. Find the stresses in the members and the reactions at \(D\) and \(E\). There is a frictionless pin joint at each corner, \(D, D', E, E'\) included. % Description of diagram: A symmetrical roof truss structure. % Base points are A, B, C, B', A'. % From left to right: A, B, C, B', A'. C is the center point. % Supports are at D and E, which are vertices of an inner square DEE'D'. % The overall structure is symmetrical about the vertical line through C. % Outer points A and F are connected to D and E respectively. % From A, member AF goes up and right. Angle FAB is 60 degrees. % From A, member AB is horizontal. % From F, member FE goes down and right. Angle at F in triangle F E G is 30 degrees. G is a point above E. % The structure is mirrored on the right side with A', F', E', D'. % Loads: 1 Ton down at A, 1 Ton down at F, 1/2 Ton down at B, 1/2 Ton down at C, 1/2 Ton down at B', 1 Ton down at F', 1 Ton down at A'. % Inner square DEE'D' has side length 3. % Horizontal distance from E to E' is 4. % The points G and G' are above E and E', on the horizontal line FF'. The distance GG' is not given, but EE' is 4' and DE is 3'. % From the diagram, it seems DEE'D' is a square of side 3'. And the horizontal distance between supports D and E is not given, but the horizontal distance EE' is 4'. This seems contradictory. Re-reading the text "The square DEE'D'". Let's assume DEE'D' is a square. Dimensions are marked on the diagram: DE = 3, EE' = 4, E'D' = 3, D'D = 4. So it's a rectangle, not a square. The problem states "The square DEE'D'". This is a contradiction in the problem statement itself. Let's assume the diagram is correct and it is a rectangle. % Angles: Angle ABD = 60 deg, Angle BDC = 60 deg, Angle CD'B' = 60 deg, Angle D'B'A' = 60 deg. % Angle GFE = 30 deg. Vertical line from G down to horizontal line DE has length 3'.
Show that two parabolas can in general be drawn through four given points, no three of which are collinear, and that the parabolas are real and distinct, if and only if each point lies outside the triangle formed by the other three. Examine the case when three of the points are collinear.
Defining the curvature of a plane curve at any point as the limit of \(\delta\psi/\delta s\) when \(\delta s\) tends to zero, obtain expressions for the coordinates of the centre and the radius of a circle which touches a given curve \(x = \phi_1(t), y = \phi_2(t)\) at a given point \(P(t=0)\) and has the same curvature there. Prove that the centre of this circle may be identified with
The sides of a triangle are 207, 480, 417; prove that one angle is 60° and find the others.
A coil of rope of mass ½ lb. per foot length lies on the ground. One end is started from rest and is pulled up vertically with a constant acceleration of 10 ft. per sec. per sec. for 2 seconds. Find the total work which has been done by the pulling agent during the 2 seconds.
Find the least distance in which a motor-car running at 20 miles an hour can be stopped by brakes on the back axle only, if the centre of gravity of the car be 3 ft. from the ground, if the wheel base is 8 ft., and if the coefficient of friction between the tyres and the road is \(\frac{2}{3}\). Assume that \(\frac{2}{3}\) of the whole weight is carried by the back axle when the car is at rest, that the weight of the wheels may be neglected, and that the front wheel bearings are frictionless.
Obtain the equation of a conic in polar coordinates, the focus being the pole, in the form \[ r(1 + e \cos \theta) = l, \] and shew that, if \(2l = e^{-1} - e\), the equation represents for different values of \(e\) a family of confocal conics, whose foci are at unit distance apart. Prove also that if in this special equation \(r\) and \(\theta\) are replaced by \(r^2\) and \(2\theta\), the curves represented are again a family of confocal conics, the distance between the foci now being two units of length.
Define the shearing stress and bending moment of a beam and show how they are connected. Illustrate by considering the case of the beam of weight \(W\) per ft. loaded as shewn and supported at \(A\) and \(B\). Give graphs of the shearing stress and bending moment. % Description of diagram: A beam supported at points A and B. % B is to the left of A. % The beam extends to the left of B and to the right of A. % Loads from left to right: % 2W at a point 3' from the left end. % Another 2W at the left end. % Support B is 3' to the right of the first 2W load. % The distance between B and A is not specified. % To the right of A: % Load W at a point 2' from A. % Load W at the right end, 1' from the previous W load.
An observer sees an aeroplane due N. at an elevation of 8°. Two minutes later he sees it N.E. at the same elevation. It is known to be going due E., the horizontal component of its velocity being 80 miles an hour. Show that it is rising at the rate of nearly 410 feet a minute.