40 N cos 35 =49N. T 1 = 28 T y

Transcription

1 16 (a) Analyzing vertical forces where string 1 and string 2 meet, we find T 1 = 40 N cos 35 =49N (b) Looking at the horizontal forces at that point leads to T 2 = T 1 sin 35 = (49 N) sin 35 =28N (c) We denote the components of T 3 as T x (rightward) and T y (upward) Analyzing horizontal forces where string 2 and string 3 meet, we find T x = T 2 = 28 N From the vertical forces there, we conclude T y = 50 N Therefore, T 3 = Tx 2 + Ty 2 =57N (d) The angle of string 3 (measured from vertical) is ( ) ( ) θ =tan 1 Tx 28 =tan 1 =29 T y 50

2 17 The cable that goes around the lowest pulley is cable 1 and has tension T 1 = F That pulley is supported by the cable 2 (so T 2 =2T 1 =2F ) and goes around the middle pulley The middle pulley is supported by cable 3 (so T 3 =2T 2 =4F ) and goes around the top pulley The top pulley is supported by the upper cable with tension T,soT =2T 3 =8F Three cables are supporting the block(which has mass m =640 kg): T 1 + T 2 + T 3 = mg = F = mg 7 =896 N Therefore, T =8(896) = 717N

3 18 (a) All forces are vertical and all distances are measured along an axis inclined at 30 Thus, any trigonometric factor cancels out and the application of torques about the contact point (referred to in the problem) leads to F tripcep = (15 kg) ( 98m/s 2) (35 cm) (20kg) ( 98m/s 2) (15 cm) 25cm (b) Equilibrium of forces (with upwards positive) leads to F tripcep + F humer +(15kg) ( 98m/s 2) (20kg) ( 98m/s 2) =0 = N and thus to F humer = N, with the minus sign implying that it points downward

4 19 (a) Analyzing the horizontal forces (which add to zero) we find F h = F 3 =50 N (b) Equilibrium of vertical forces leads to F v = F 1 + F 2 =30N (c) Computing torques about point O, weobtain F v d = F 2 b + F 3 a = d = (10)(30) + (50)(20) 30 =13 m

5 20 (a) The sign is attached in two places: at x 1 = 100 m (measured rightward from the hinge) and at x 2 =300 m We assume the downward force due to the sign s weight is equal at these two attachment points: each being half the sign s weight of mg The angle where the cable comes into contact (also at x 2 )isθ =tan 1 (4/3) and the force exerted there is the tension T Computing torques about the hinge, we find T = 1 2 mgx mgx 2 = x 2 sin θ 1 2 (500)(98)(100) (500)(98)(300) = 408 N (300)(0800) (b) Equilibrium of horizontal forces requires the (rightward) horizontal hinge force be F x = T cos θ = 245 N (c) And equilibrium of vertical forces requires the (upward) vertical hinge force be F y = mg T sin θ = 163 N

6 21 We consider the wheel as it leaves the lower floor The floor no longer exerts a force on the wheel, and the only forces acting are the force F applied horizontally at the axle, the force of gravity mg acting vertically at the center of the wheel, and the force of the step corner, shown as the two components f h and f v If the minimum force is applied the wheel does not accelerate, so both the total force and the total torque acting on it are zero mg f h F f v h We calculate the torque around the step corner The second diagram indicates that the distance from the line of F to the corner is r h, wherer is the radius of the wheel and h is the height of the step The distance from the line of mg to the corner is r 2 +(r h) 2 = 2rh h 2 Thus F (r h) mg 2rh h 2 = 0 The solution for F is r h r 2rh h 2 F = mg r h h

7 22 (a) The problem asks for the person s pull (his force exerted on the rock) but since we are examining forces and torques on the person, we solve for the reaction force N 1 (exerted leftward on the hands by the rock) At that point, there is also an upward force of static friction on his hands f 1 which we will take to be at its maximum value µ 1 N 1 We note that equilibrium of horizontal forces requires N 1 = N 2 (the force exerted leftward on his feet); on this feet there is also an upward static friction force of magnitude µ 2 N 2 Equilibrium of vertical forces gives f 1 + f 2 mg =0 = N 1 = mg = N µ 1 + µ 2 (b) Computing torques about the point where his feet come in contact with the rock, we find mg(d + w) f 1 w N 1 h =0 = h = mg(d + w) µ 1N 1 w N 1 =088 m (c) Both intuitively and mathematically (since both coefficients are in the denominator) we see from part (a) that N 1 would increase in such a case As for part (b), it helps to plug part (a) into part (b) and simplify: h =(d + w)µ 2 + dµ 1 from which it becomes apparent that h should decrease if the coefficients decrease

8 23 The beam is in equilibrium: the sum of the forces and the sum of the torques acting on it each vanish As we see in the figure, the beam makes an angle of 60 with the vertical and the wire makes an angle of 30 with the vertical (a) We calculate the torques around the hinge Their sum is TLsin 30 W (L/2) sin 60 =0 HereW is the force of gravity acting at the center of the beam, and T is the tension force of the wire We solve for the tension: W sin 60 (222 N) sin 60 T = = 2 sin 30 2 sin 30 = 1923 N (b) Let F h be the horizontal component of the force exerted by the hinge and take it to be positive if the force is outward from the wall Then, the vanishing of the horizontal component of the net force on the beam yields F h T sin 30 =0or F h = T sin 30 = (1923 N) sin 30 =961 N (c) Let F v be the vertical component of the force exerted by the hinge and take it to be positive if it is upward Then, the vanishing of the vertical component of the net force on the beam yields F v + T cos 30 W =0or F v = W T cos 30 = 222 N (1923 N) cos 30 =655 N

9 24 (a) The top brick s center of mass cannot be further (to the right) with respect to the brick below it (brick 2) than L/2; otherwise, its center of gravity is past any point of support and it will fall So a 1 = L/2 inthemaximumcase (b) With brick 1 (the top brick) in the maximum situation, then the combined center of mass of brick 1 and brick 2 is halfway between the middle of brick 2 and its right edge That point (the combined com) must be supported, so in the maximum case, it is just above the right edge of brick 3 Thus, a 2 = L/4 (c) Now the total center of mass of bricks 1, 2 and 3 is one-third of the way between the middle of brick 3 and its right edge, as shown by this calculation: 2m(0) + m( L/2) x com = = L 3m 6 where the origin is at the right edge of brick 3 This point is above the right edge of brick 4 in the maximum case, so a 3 = L/6 (d) A similar calculation shows that a 4 = L/8 (e) We find h = 4 i=1 a i =25L/24 x com = 3m(0) + m( L/2) 4m = L 8

10 25 (a) We note that the angle θ between the cable and the strut is =15 The angle φ between the strut and any vertical force (like the weights in the problem) is =45 Denoting M = 225 kg and m =450 kg, and l as the length of the boom, we compute torques about the hinge and find T = Mglsin φ + mg ( l 2) sin φ l sin θ The unknown length l cancels out and we obtain T = N (b) Since the cable is at 30 from horizontal, then horizontal equilibrium of forces requires that the horizontal hinge force be F x = T cos 30 = N (c) And vertical equilibrium of forces gives the vertical hinge force component: F y = Mg + mg + T sin 30 = N

11 26 (a) The problem states that each hinge supports half the door s weight, so each vertical hinge force component is F y = mg/2 = N (b) Computing torques about the top hinge, we find the horizontal hinge force component (at the bottom hinge) is F h = (27 kg) ( 98m/s 2)( ) 091 m 2 =80N 21m 2(030m) Equilibrium of horizontal forces demands that the horizontal component of the top hinge force has the same magnitude (though opposite direction)

12 27 The bar is in equilibrium, so the forces and the torques acting on it each sum to zero Let T l be the tension force of the left-hand cord, T r be the tension force of the right-hand cord, and m be the mass of the bar The equations for equilibrium are: vertical force components T l cos θ + T r cos φ mg =0 horizontal force components T l sin θ + T r sin φ =0 torques mgx T r L cos φ =0 The origin was chosen to be at the left end of the bar for purposes of calculating the torque The unknown quantities are T l, T r,andx We want to eliminate T l and T r,thensolveforx The second equation yields T l = T r sin φ/ sin θ and when this is substituted into the first and solved for T r the result is T r = mg sin θ/(sin φ cos θ +cosφsin θ) This expression is substituted into the third equation and the result is solved for x: sin θ cos φ sin θ cos φ x = L = L sin φ cos θ +cosφsin θ sin(θ + φ) The last form was obtained using the trigonometric identity sin(a + B) = sin A cos B +cosasin B For the special case of this problem θ + φ =90 and sin(θ + φ) =1 Thus, x = L sin θ cos φ =(610m) sin 369 cos 531 =220m

13 28 (a) Computing torques about the hinge, we find the tension in the wire: TLsin θ Wx =0 = T = Wx L sin θ (b) The horizontal component of the tension is T cos θ, so equilibrium of horizontal forces requires that the horizontal component of the hinge force is ( ) Wx F x = cos θ = Wx L sin θ L tan θ (c) The vertical component of the tension is T sin θ, so equilibrium of vertical forces requires that the vertical component of the hinge force is ( ) Wx ( F y = W sin θ = W 1 x ) L sin θ L

14 29 The diagram on the right shows the forces acting on the plank Since the roller is frictionless the force it exerts is normal to the plank and makes the angle θ with the vertical Its magnitude is designated F W is the force of gravity; this force acts at the center of the plank, a distance L/2 from the point where the plank touches the floor N is the normal force of the floor and f is the force of friction The distance from the foot of the plank to the wall is denoted by d This quantity is not given directly but it can be computed using d = h/ tan θ The equations of equilibrium are: N θ F f W θ d horizontal force components F sin θ f =0 vertical force components F cos θ W + N =0 torques Nd fh W ( d L 2 cos θ) =0 h The point of contact between the plank and the roller was used as the origin for writing the torque equation When θ =70 the plank just begins to slip and f = µ s N,whereµ s is the coefficient of static friction We want to use the equations of equilibrium to compute N and f for θ =70,thenuseµ s = f/n to compute the coefficient of friction The second equation gives F =(W N)/ cos θ and this is substituted into the first to obtain f = (W N) sin θ/ cos θ =(W N)tanθ This is substituted into the third equation and the result is solved for N: d (L/2) cos θ + h tan θ N = W d + h tan θ Now replace d with h/ tan θ and multiply both numerator and denominator by tan θ The result is N = h(1 + tan2 θ) (L/2) sin θ h(1 + tan 2 θ) We use the trigonometric identity 1 + tan 2 θ =1/ cos 2 θ and multiply both numerator and denominator bycos 2 θ to obtain ( N = W 1 L ) 2h cos2 θ sin θ W Now we use this expression for N in f =(W N)tanθ to find the friction: f = WL 2h sin2 θ cos θ We substitute these expressions for f and N into µ s = f/n and obtain µ s = L sin2 θ cos θ 2h L sin θ cos 2 θ Evaluating this expression for θ =70,weobtain µ s = (61 m) sin 2 70 cos 70 2(305 m) (61 m) sin 70 cos 2 70 =034

15 30 (a) Computing torques about point A, we find T max L sin θ = Wx max + W b ( L 2 ) We solve for the maximum distance: x max = T max sin θ W b 2 W (b) Equilibrium of horizontal forces gives (c) And equilibrium of vertical forces gives L = 500 sin (30) = 15 m 300 F x = T max cos θ = 433 N F y = W + W b T max sin θ = 250 N

16 31 The diagrams to the right show the forces on the two sides of the ladder, separated F A and F E F v are the forces of the floor on the two feet, T is F h F h the tension force of the tie rod, W is the force of the man (equal to his weight), F h is the horizontal F v component of the force exerted by one side of the ladder on the other, and F v is the vertical component of that force Note that the forces exerted T T by the floor are normal to the floor since the floor F A W F E is frictionless Also note that the force of the left side on the right and the force of the right side θ θ on the left are equal in magnitude and opposite in direction Since the ladder is in equilibrium, the vertical components of the forces on the left side of the ladder must sum to zero: F v + F A W = 0 The horizontal components must sum to zero: T F h =0 The torques must also sum to zero We take the origin to be at the hinge and let L be the length of a ladder side Then F A L cos θ W (L/4) cos θ T (L/2) sin θ =0 Herewerecognizethatthemanisone-fourth the length of the ladder side from the top and the tie rod is at the midpoint of the side The analogous equations for the right side are F E F v =0,F h T =0,andF E L cos θ T (L/2) sin θ =0 There are 5 different equations: F v + F A W =0, T F h =0 F A L cos θ W (L/4) cos θ T (L/2) sin θ =0 F E F v =0 F E L cos θ T (L/2) sin θ =0 The unknown quantities are F A, F E, F v, F h,andt (a) First we solve for T by systematically eliminating the other unknowns The first equation gives F A = W F v and the fourth gives F v = F E We use these to substitute into the remaining three equations to obtain T F h =0 WLcos θ F E L cos θ W (L/4) cos θ T (L/2) sin θ =0 F E L cos θ T (L/2) sin θ =0 The last of these gives F E = T sin θ/2cosθ =(T/2) tan θ We substitute this expression into the second equation and solve for T The result is T = 3W 4tanθ To find tan θ, we consider the right triangle formed by the upper half of one side of the ladder, half the tie rod, and the vertical line from the hinge to the tie rod The lower side of the triangle has alengthof0381 m, the hypotenuse has a length of 122 m, and the vertical side has a length of (122 m)2 (0381 m) 2 =116 m This means tan θ =(116 m)/(0381 m) = 304 Thus, 3(854 N) T = 4(304) = 211 N (b) We now solve for F A SinceF v = F E and F E = T sin θ/2cosθ, F v =3W/8 We substitute this into F v + F A W = 0 and solve for F A We find F A = W F v = W 3W/8 =5W/8 = 5(884 N)/8 = 534 N (c) We have already obtained an expression for F E : F E =3W/8 Evaluating it, we get F E = 320 N

17 32 The phrase loosely bolted means that there is no torque exerted by the bolt at that point (where A connects with B) The force exerted on A at the hinge has x and y components F x and F y Theforce exerted on A at the bolt has components G x and G y and those exerted on B are simply G x and G y by Newton s third law The force exerted on B at its hinge has components H x and H y Ifahorizontal force is positive, it points rightward, and if a vertical force is positive it points upward (a) We consider the combined A B system, which has a combined weight of Mg where M = 122 kg and the line of action of that downward force of gravity is x =120mfromthewall Thevertical distance between the hinges is y =180 m We compute torques about the bottom hinge and find F x = Mgx = 797 N y If we examine the forces on A alone and compute torques about the bolt, we instead find F y = m Agx = 265 N l where m A =540 kgandl =240 m (the length of beam A) (b) Equilibrium of horizontal and vertical forces on beam A readily yields G x = F x = 797 N and G y = m A g F y = 265 N (c) Considering again the combined A B system, equilibrium of horizontal and vertical forces readily yields H x = F x = 797 N and H y = Mg F y = 931 N (d) As mentioned above, Newton s third law (and the results from part (b)) immediately provide G x = 797 N and G y = 265 N for the force components acting on B at the bolt

18 33 We examine the box when it is about to tip Since it will rotate about the lower right edge, that is where the normal force of the floor is exerted This force is labeled N on the diagram to the right The force of friction is denoted by f, the applied force by F, and the force of gravity by W Note that the force of gravity is applied at the center of the box When the minimum force is applied the box does not accelerate, so the sum of the horizontal force components vanishes: F f = 0, the sum of the vertical force components vanishes: N W =0,andthe sum of the torques vanishes: FL WL/2=0 HereL is the length of a side of the box and the origin was chosen to be at the lower right edge F f W N (a) From the torque equation, we find F = W 2 = 890 N 2 = 445 N (b) The coefficient of static friction must be large enough that the box does not slip The box is on the verge of slipping if µ s = f/n According to the equations of equilibrium N = W = 890 N and f = F = 445 N, so µ s = 445 N 890 N =050 (c) The box can be rolled with a smaller applied force if the force points upward as well as to the right Let θ be the angle the force makes with the horizontal The torque equation then becomes FLcos θ + FLsin θ WL/2 = 0, with the solution F = W 2(cos θ + sin θ) F θ f W N We want cos θ + sin θ to have the largest possible value This occurs if θ =45, a result we can prove by setting the derivative of cos θ + sin θ equal to zero and solving for θ The minimum force needed is W F = 4 cos 45 = 890 N 4 cos 45 = 315 N

19 34 We locate the origin of the x axis at the edge of the table and choose rightwards positive The criterion (in part (a)) is that the center of mass of the block above another must be no further than the edge of the one below; the criterion in part (b) is more subtle and is discussed below Since the edge of the table corresponds to x =0 then the total center of mass of the blocks must be zero (a) We treat this as three items: one on the upper left (composed of two bricks, one directly on top of the other) of mass 2m whose center is above the left edge of the bottom brick; a single brick at the upper right of mass m which necessarily has its center over the right edge of the bottom brick (so a 1 = L/2 trivially); and, the bottom brick of mass m The total center of mass is (2m)(a 2 L)+ma 2 + m(a 2 L/2) =0 4m which leads to a 2 =5L/8 Consequently, h = a 2 + a 1 =9L/8 (b) We have four bricks (each of mass m) where the center of mass of the top and the center of mass of the bottom one have the same value x cm = b 2 L/2 The middle layer consists of two bricks, and we note that it is possible for each of their centers of mass to be beyond the respective edges of the bottom one! This is due to the fact that the top brick is exerting downward forces (each equal to half its weight) on the middle blocks and in the extreme case, this may be thought of as a pair of concentrated forces exerted at the innermost edges of the middle bricks Also, in the extreme case, the support force (upward) exerted on a middle block (by the bottom one) F top may be thought of as a concentrated force located at the edge of the bottom block (which is the point about which we compute torques, in the following) If (as indicatedinoursketch,wheref top has magnitude mg/2) we consider equilibrium of m g torques on the rightmost brick, we obtain mg (b 1 12 ) L = mg 2 (L b 1) F bottom which leads to b 1 =2L/3 Once we conclude from symmetry that b 2 = L/2 then we also arrive at h = b 2 + b 1 =7L/6

20 35 The force diagram shown on the right depicts the situation just before the crate tips, when the normal force acts at the front edge However, it may also be used to calculate the angle for which the crate begins to slide W is the force of gravity on the crate, N is the normal force of the plane on the crate, and f is the force of friction We take the x axis to be down the plane and the y axis to be in the direction of the normal force We assume the acceleration is zero but the crate is on the verge of sliding f W N θ (a) The x and y components of Newton s second law are W sin θ f =0 and N W cos θ =0 respectively The y equation gives N = W cos θ Since the crate is about to slide f = µ s N = µ s W cos θ, whereµ s is the coefficient of static friction We substitute into the x equation and find W sin θ µ s W cos θ =0 = tan θ = µ s This leads to θ =tan 1 µ s =tan = 310 In developing an expression for the total torque about the center of mass when the crate is about to tip, we find that the normal force and the force of friction act at the front edge The torque associated with the force of friction tends to turn the crate clockwise and has magnitude fh,where h is the perpendicular distance from the bottom of the crate to the center of gravity The torque associated with the normal force tends to turn the crate counterclockwise and has magnitude Nl/2, where l is the length of a edge Since the total torque vanishes, fh = Nl/2 When the crate is about to tip, the acceleration of the center of gravity vanishes, so f = W sin θ and N = W cos θ Substituting these expressions into the torque equation, we obtain θ =tan 1 l 12m 2h =tan 1 2(090 m) =337 As θ is increased from zero the crate slides before it tips It starts to slide when θ =310 (b) The analysis is the same The crate begins to slide when θ =tan 1 µ s =tan = 350 and begins to tip when θ =337 Thus, it tips first as the angle is increased Tipping begins at θ =337

21 36 (a) The Young s modulus is given by E = stress = slope of the stress-strain curve strain = N/m 2 = N/m (b) Since the linear range of the curve extends to about N/m 2, this is approximately the yield strength for the material