APPLYING NEWTON S LAWS

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Evan Horn
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1 APPLYING NEWTON S LAWS 5 igible mass. Let T r be the tension in the rope and let T c be the tension in the chain. EXECUTE: (a) The free-bod diagram for each weight is the same and is given in Figure 5.1a. Σ F = ma gives Tr = w= 5. 0 N. (b) The free-bod diagram for the pulle is given in Figure 5.1b. Tc = Tr = 500. N. EVALUATE: The tension is the same at all points along the rope IDENTIFY SET UP: T ake + upw ard. The pull e has negl Figure 5.1a, b 5.6. IDENTIFY: Appl Newton s first law to the wrecking ball. Each cable eerts a force on the ball, directed along the cable. SET UP: The force diagram for the wrecking ball is sketched in Figure 5.6. Figure 5.6

2 EXECUTE: (a) Σ F = ma TB cos40 mg = 0 mg (4090 kg)(9. 80 m/s ) 4 T B = = = N cos40 cos40 (b) Σ F = ma B T sin 40 T = 0 T A B EVALUATE: A = T sin40 = N 4 If the angle 40 is replaced b 0 (cable B is vertical), then TB = mg and T = IDENTIFY: Appl Σ F = ma to the object and to the knot where the cords are joined. SET UP: Let + be upward and + be to the right. EXECUTE: (a) TC = w, TA sin30 + TBsin45 = TC = w, and TAcos30 TBcos45 = 0. Since w sin 45 = cos45, adding the last two equations gives TA(cos30 + sin30 ) = w, and so TA = = 073. w cos30 Then, TB = TA = w. cos45 (b) Similar to part (a), TC = w, TAcos60 + TBsin45 = w, and TAsin60 TBcos45 = 0. w sin60 Adding these two equations, TA = = 73. w, and TB = TA = 335. w. (sin 60 cos60 ) cos45 EVALUATE: In part (a), TA+ TB > w since onl the vertical components of T A and T B hold the object against gravit. In part (b), since T A has a downward component T B is greater than w IDENTIFY: Appl Newton s first law to the hanging weight and to each knot. The tension force at each end of a string is the same. (a) Let the tensions in the three strings be T, Tʹ, and Tʹ ʹ, as shown in Figure 5.10a. A Figure 5.10a SET UP: The free-bod diagram for the block is given in Figure 5.10b. EXECUTE: Σ F =0 Tʹ w= 0 Tʹ = w= N Figure 5.10b SET UP: The free-bod diagram for the lower knot is given in Figure 5.10c.

3 EXECUTE: Σ F = 0 T sin 45 Tʹ = 0 Tʹ N T = = = N sin 45 sin 45 Figure 5.10c (b) Appl Σ F = 0 to the force diagram for the lower knot: Σ F = 0 F = T cos45 = (84. 9 N)cos45 = N SET UP: The free-bod diagram for the upper knot is given in Figure 5.10d. EXECUTE: Σ F = 0 Tcos45 F1 = 0 F 1 = (84. 9 N)cos45 F 1 = N Figure 5.10d Note that F 1 = F. EVALUATE: Appling Σ F = 0 to the upper knot gives Tʹ ʹ = Tsin 45 = N = w. If we treat the whole sstem as a single object, the force diagram is given in Figure 5.10e. Σ = 0 gives F = F 1, which checks F Σ F = 0 gives Tʹ ʹ = w, which checks Figure 5.10e 5.1. IDENTIFY: Appl Newton s second law to the rocket plus its contents and to the power suppl. Both the rocket and the power suppl have the same acceleration. SET UP: The free-bod diagrams for the rocket and for the power suppl are given in Figures 5.1a and b. Since the highest altitude of the rocket is 10 m, it is near to the surface of the earth and there is a downward gravit force on each object. Let + be upward, since that is the direction of the acceleration. The power suppl has mass m ps = (15. 5 N)/(9. 80 m/s ) = kg. EXECUTE: (a) Σ F = ma applied to the rocket gives F mrg = ma r. F mr g 170 N (15 kg)(9. 80 m/s ) a = = = 396. m/s. m 15 kg (b) Σ = r F ma applied to the power suppl gives n m ps g = m ps a. n= m ps ( g+ a ) = (1. 58 kg)(9. 80 m/s m/s ) = 1. 7 N. EVALUATE: The acceleration is constant while the thrust is constant and the normal force is constant while the acceleration is constant. The altitude of 10 m is not used in the calculation.

4 Figure IDENTIFY: Appl Σ F = ma to the load of bricks and to the counterweight. The tension is the same at each end of the rope. The rope pulls up with the same force ( T ) on the bricks and on the counterweight. The counterweight accelerates downward and the bricks accelerate upward; these accelerations have the same magnitude. (a) SET UP: The free-bod diagrams for the bricks and counterweight are given in Figure Figure 5.15 (b) EXECUTE: Appl Σ F = ma to each object. The acceleration magnitude is the same for the two objects. For the bricks take + to be upward since a r for the bricks is upward. For the counterweight take + to be downward since a r is downward. bricks: Σ F = ma T mg = ma 1 1 counterweight: Σ F = ma mg T= ma Add these two equations to eliminate T: ( m m ) g = ( m + m ) a 1 1 m m kg kg (9 80 m/s ) 96 m/s a= g =. =. m1+ m kg kg (c) T mg 1 = ma 1 gives T = m 1 ( a+ g ) = (150. kg)(96. m/s m/s ) = 191 N As a check, calculate T using the other equation. mg T= ma gives T = m ( g a ) = 8.0kg(9.80m/s.96m/s ) = 191 N, which checks. EVALUATE: The tension is 1.30 times the weight of the bricks; this causes the bricks to accelerate upward. The tension is times the weight of the counterweight; this causes the counterweight to accelerate downward. If m 1 = m, a = 0 and T = mg 1 = m g. In this special case the objects don t move. If m 1 = 0, a= g and T = 0; in this special case the counterweight is in free fall. Our general result is correct in these two special cases.

5 5.0. IDENTIFY: Appl Σ F = ma to the composite object of elevator plus student ( m tot = 850 kg) and also to the student ( w = 550 N). The elevator and the student have the same acceleration. SET UP: Let + be upward. The free-bod diagrams for the composite object and for the student are given in Figures 5.0a and b. T is the tension in the cable and n is the scale reading, the normal force the scale eerts on the student. The mass of the student is m = w/g = kg. EXECUTE: (a) Σ F = ma applied to the student gives n mg = ma. n mg 450 N 550 N a = = = 178. m/s. The elevator has a downward acceleration of 178. m/s. m kg 670 N 550 N (b) a = = 14. m/s kg (c) n = 0 means a = g. The student should worr; the elevator is in free fall. (d) Σ F = ma applied to the composite object gives T mtotg mtot a. = T = mtot ( a + g ). In part (a), T = (850 kg)( m/s m/s ) = 680 N. In part (c), a = g and T = 0. EVALUATE: In part (b), T = (850 kg)(. 14 m/s m/s ) = 10, 150 N. The weight of the composite object is 8330 N. When the acceleration is upward the tension is greater than the weight and when the acceleration is downward the tension is less than the weight. Figure 5.0a, b 5.7. (a) IDENTIFY: Constant speed implies a = 0. Appl Newton s first law to the bo. The friction force is directed opposite to the motion of the bo. SET UP: Consider the free-bod diagram for the bo, given in Figure 5.7a. Let F r be the horizontal force applied b the worker. The friction is kinetic friction since the bo is sliding along the surface. EXECUTE: Σ F = ma n mg = 0 n= mg so fk = µ kn= µ kmg Figure 5.7a Σ F = ma F f k = 0 F = fk = µ k mg= (0. 0)(11. kg)(9. 80 m/s ) = N (b) IDENTIFY: Now the onl horizontal force on the bo is the kinetic friction force. Appl Newton s second law to the bo to calculate its acceleration. Once we have the acceleration, we can find the distance using a constant acceleration equation. The friction force is fk = µ k mg, just as in part (a). SET UP: The free-bod diagram is sketched in Figure 5.7b.

6 EXECUTE: Σ F = ma f = ma k k µ mg = ma a = µ k g= (0. 0)(9. 80 m/s ) = m/s Figure 5.7b Use the constant acceleration equations to find the distance the bo travels: v = 0, v 0 = m/s, a = 196. m/s, 0 =? v = v 0 + a( 0 ) v v0 0 (3. 50 m/s) 0 = = = 31. m a ( m/s ) EVALUATE: The normal force is the component of force eerted b a surface perpendicular to the surface. Its magnitude is determined b Σ F =ma. In this case n and mg are the onl vertical forces and a = 0, so n= mg. Also note that f k and n are proportional in magnitude but perpendicular in direction IDENTIFY: Constant speed means zero acceleration for each block. If the block is moving, the friction force the tabletop eerts on it is kinetic friction. Appl Σ F = ma to each block. SET UP: The free-bod diagrams and choice of coordinates for each block are given b Figure m A = 4.59 kg and m B =.55 kg. EXECUTE: (a) Σ F = ma with a = 0 applied to block B gives mg B T= 0 and T = 5.0 N. Σ F = ma with a = 0 applied to block A gives T fk = 0 and f k = 5.0 N. na = mag = 45.0 N and fk 5.0 N µ k = = = na 45.0 N (b) Now let A be block A plus the cat, so m A = 9.18 kg. n A = 90.0 N and fk = µ k n= (0.556)(90.0 N) = 50.0 N. F = mafor A gives T fk = maa. F = ma for block B gives mbg T = mba. a for A equals a for B, so adding the two equations gives mbg fk = ( ma+ mb) a and mbg fk 5.0 N 50.0 N a = = =.13 m/s. The acceleration is upward and block B slows down. ma+ mb 9.18 kg +.55 kg EVALUATE: The equation mbg fk = ( ma+ mb) a has a simple interpretation. If both blocks are considered together then there are two eternal forces: mg B that acts to move the sstem one wa and f k that acts oppositel. The net force of mg B fk must accelerate a total mass of ma+ mb. Figure (a) IDENTIFY: Appl Σ F = ma to the crate. Constant v implies a = 0. Crate moving sas that the friction is kinetic friction. The target variable is the magnitude of the force applied b the woman.

7 SET UP: The free-bod diagram for the crate is sketched in Figure EXECUTE: Σ F = ma n mg Fsinθ = 0 n= mg + Fsinθ f µ n µ mg µ F θ k = k = k + k sin Figure 5.39 Σ F = ma Fcosθ fk = 0 Fcosθ µ kmg µ kfsinθ = 0 F(cosθ µ ksin θ ) = µ kmg µ kmg F = cosθ µ k sinθ (b) IDENTIFY and SET UP: start the crate moving means the same force diagram as in part (a), ecept that µ k is replaced b µ s. Thus µ smg F =. cosθ µ sinθ s cosθ 1 EXECUTE: F if cosθ µ s sinθ = 0. This gives µ s = =. sinθ tanθ EVALUATE: F r has a downward component so n> mg. If θ = 0 (woman pushes horizontall), n= mg and F = f = mg k µ k IDENTIFY: Appl Σ F = ma to the ball. At the terminal speed, f = mg. SET UP: The fluid resistance is directed opposite to the velocit of the object. At half the terminal speed, the magnitude of the frictional force is one-fourth the weight. EXECUTE: (a) If the ball is moving up, the frictional force is down, so the magnitude of the net force is (5/4)w and the acceleration is (5/4) g, down. (b) While moving down, the frictional force is up, and the magnitude of the net force is (3/4)w and the acceleration is (3/4) g, down. EVALUATE: The frictional force is less than mg in each case and in each case the net force is downward and the acceleration is downward IDENTIFY and SET UP: Appl Eq. (5.13). EXECUTE: (a) Solving for D in terms of t, (b) v t = mg (45 kg)(9.80 m/s ) 4 m/s. D = (0.5 kg/m) = EVALUATE: v mg D v t (4 m/s) (80 kg)(9.80 m/s ) = = = 0.44 kg/m. Terminal speed depends on the mass of the falling object IDENTIFY: The acceleration of the car at the top and bottom is toward the center of the circle, and Newton s second law applies to it. SET UP: Two forces are acting on the car, gravit and the normal force. At point B (the top), both forces are toward the center of the circle, so Newton s second law gives mg + nb = ma. At point A (the bottom), gravit is downward but the normal force is upward, so na mg = ma.

8 EXECUTE: Solving the equation at B for the acceleration gives mg + n B.. +. Solving the equation at A for the normal force gives (0 800 kg)(9 8 m/s ) 6 00 N a = = = m/s. m kg na = m( g+ a) = (0800. kg)(98. m/s m/s ) = 17. N. EVALUATE: The normal force at the bottom is greater than at the top because it must balance the weight in addition to accelerate the car toward the center of its track IDENTIFY: Since the car travels in an arc of a circle, it has acceleration arad = v / R, directed toward the center of the arc. The onl horizontal force on the car is the static friction force eerted b the roadwa. To calculate the minimum coefficient of friction that is required, set the static friction force equal to its maimum value, f s = µ s n. Friction is static friction because the car is not sliding in the radial direction. SET UP: The free-bod diagram for the car is given in Figure The diagram assumes the center of the curve is to the left of the car. s. v EXECUTE: (a) Σ F = ma gives n= mg. Σ F = ma gives µ n= m R v (5. 0 m/s) µ s = = = 090. gr (9. 80 m/s )(0 m) v (b) Rg constant, µ = = so v1 v µ s µ s1 =. v v 1 s µ s1 µ s µ s1 µ s1 EVALUATE: /3 = = (5. 0 m/s) = m/s. v µ smg = m and R A smaller coefficient of friction means a smaller maimum friction force, a smaller possible acceleration and therefore a smaller speed. Figure π R IDENTIFY: The acceleration due to circular motion is arad =. T SET UP: R = 400 m. 1/T is the number of revolutions per second. EXECUTE: (a) Setting arad = g and solving for the period T gives R 400 m T = π = π = 40.1 s, g 9.80 m/s so the number of revolutions per minute is (60 s/min)/(40.1 s) = 1.5 rev/min. (b) The lower acceleration corresponds to a longer period, and hence a loweotation rate, b a factor of the square root of the ratio of the accelerations, T ʹ = (1.5 rev/min) 3.70/ 9.8 = 0.9 rev/min. EVALUATE: v Ra Rg In part (a) the tangential speed of a point at the rim is given b a = rad = = 6.6 m/s; the space station is rotating rapidl. v rad, = so R 5.59.IDENTIFY: Appl Newton s first law to the ball. The force of the wall on the ball and the force of the ball on the wall are related b Newton s third law.

9 SET UP: The forces on the ball are its weight, the tension in the wire, and the normal force applied b the wall. To 16.0 cm calculate the angle φ that the wire makes with the wall, use Figure 5.59a. sinφ = and φ = cm EXECUTE: (a) The free-bod diagram is shown in Figure 5.59b. Use the and coordinates shown in the w (45.0 kg)(9.80 m/s ) figure. Σ F = 0 gives Tcosφ w= 0 and T = = = 470 N cosφ cos0.35 (b) Σ F = 0 gives Tsinφ n= 0. n = (470 N)sin0.35 = 163 N. B Newton s third law, the force the ball eerts on the wall is 163 N, directed to the right. w EVALUATE: n= sinφ = wtan φ. As the angle φ decreases (b increasing the length of the wire), cosφ T decreases and n decreases. Figure 5.59a, b IDENTIFY: Appl Newton s first law to the ball. Treat the ball as a particle. SET UP: The forces on the ball are gravit, the tension in the wire and the normal force eerted b the surface. The normal force is perpendicular to the surface of the ramp. Use and aes that are horizontal and vertical. EXECUTE: (a) The free-bod diagram for the ball is given in Figure The normal force has been replaced b its and components. mg (b) Σ F = 0 gives ncos35.0 w= 0 and n= = 1. mg. cos35.0 (c) Σ F = 0 gives T nsin35.0 = 0 and T = (1. mg)sin35.0 = mg. EVALUATE: Note that the normal force is greater than the weight, and increases without limit as the angle of the ramp increases toward 90. The tension in the wire is wtan φ, where φ is the angle of the ramp and T also increases without limit as φ 90.

10 Figure IDENTIFY: This is a sstem having constant acceleration, so we can use the standard kinematics formulas as well as Newton s second law to find the unknown mass m. SET UP: Newton s second law applies to each block. The standard kinematics formulas can be used to find the acceleration because the acceleration is constant. The normal force on m 1 is mg 1 cos α, so the force of friction on it is f k = µ k mg 1 cos α. EXECUTE: Standard kinematics gives the acceleration of the sstem to be ( 0) (1. 0 m) a = = =. 667 m/s. For m 1, n= mg 1 cosα = N, so the friction force on m 1 is t (3. 00 s) f k = (0. 40)( N) = N. Appling Newton s second law to m 1 gives T fk mg 1 sin α = ma 1, where T is the tension in the cord. Solving for T gives T = fk + mg 1 sinα + ma 1 = N N N = N. Newton s second law for m gives mg T = ma, so T N m = = = kg. g a 9. 8 m/s. 667 m/s EVALUATE: This problem is similar to Problem 5.67, ecept for the sloped surface. As in that problem, we could treat these blocks as a two-block sstem. Newton s second law would then give mg mgsinα µ mgcos α = ( m+ m) a, which gives the same result as above. 1 k IDENTIFY: Appl Σ F = ma to the instrument and calculate the acceleration. Then use constant acceleration equations to describe the motion. SET UP: The free-bod diagram for the instrument is given in Figure The instrument has mass m= w/ g = kg. EXECUTE: (a) Adding the forces on the instrument, we have Σ F = ma, which gives T mg = ma and T mg a = = m/s. v0 = 0, v = 330 m/s, a = m/s, t =? Then v = v0 + at gives t = s. m Consider forces on the rocket; rocket has the same a. Let F be the thrust of the rocket engines. F mg = ma and 5 F = m( g+ a ) = (5,000 kg)(980. m/s + 196m/s. ) = N. (b) 1 0 v0t a t 0 EVALUATE: = + gives = 770 m. The rocket and instrument have the same acceleration. The tension in the wire is over twice the weight of the instrument and the upward acceleration is greater than g. Figure IDENTIFY: Appl Σ F = ma to the point where the three wires join and also to one of the balls. B smmetr the tension in each of the 35.0 cm wires is the same. SET UP: The geometr of the situation is sketched in Figure 5.87a. The angle φ that each wire makes with 1. 5 cm the vertical is given b sinφ = and φ = Let T A be the tension in the vertical wire and let T B be cm the tension in each of the other two wires. Neglect the weight of the wires. The free-bod diagram for the lefthand ball is given in Figure 5.87b and for the point where the wires join in Figure 5.87c. n is the force one ball eerts on the other.

11 EXECUTE: (a) Σ F = ma applied to the ball gives TB cosφ mg = 0. mg (15. 0 kg)(9. 80 m/s ) TB = = = 15 N. Then Σ F = ma applied in Figure 5.87c gives TA TBcosφ = 0 cosφ cos15. 6 and TA = (15 N)cosφ = 94 N. (b) Σ F = ma applied to the ball gives n TB sinφ = 0 and n = (15 N)sin15. 6 = N. EVALUATE: T equals the total weight of the two balls. A Figure 5.87a c IDENTIFY: Appl Σ F = ma to the automobile. v SET UP: The correct banking angle is for zero friction and is given b 0 tan β =, as derived in Eample gr 5.. Use coordinates that are vertical and horizontal, since the acceleration is horizontal. EXECUTE: For speeds larger than v 0, a frictional force is needed to keep the car from skidding. In this case, the inward force will consist of a part due to the normal force n and the friction force f; n sinβ + f cos β = marad. The normal and friction forces both have vertical components; since there is no ( 15. v ) 0 vertical acceleration, n cos β f sinβ = mg. Using f = µ sn and arad = v = = 5. gtan β, these two R R relations become nsin β + µ sncos β =. 5 mgtan β and ncos β µ snsin β = mg. Dividing to cancel n gives sin β + µ s cos β 1. 5 sin β cosβ =. 5 tan β. Solving for µ s and simplifing ields µ s =. Using cos β µ sin β sin β s (0 m/s) β = arctan = (9.80 m/s )(10 m) gives µ s = EVALUATE: If µ s is insufficient, the car skids awa from the center of curvature of the roadwa, so the friction is inward IDENTIFY: Appl Σ F = ma to each block. The tension in the string is the same at both ends. If T < w for a block, that block remains at rest. SET UP: In all cases, the tension in the string will be half of F. EXECUTE: (a) F / = 6 N, which is insufficient to raise either block; a1= a = 0. (b) F / = 147 N. The larger block (of weight 196 N) will not move, so a 1 = 0, but the smaller block, of weight 49 N 98 N, has a net upward force of 49 N applied to it, and so will accelerate upward with a = = 4.9 m/s kg

12 (c) F / = 1 N, so the net upward force on block A is 16 N and that on block B is 114 N, so 16 N 114 N a 1 = = 0.8 m/s and a = = 11.4 m/s. 0.0 kg 10.0 kg EVALUATE: The two blocks need not have accelerations with the same magnitudes.

Figure 5.1a, b IDENTIFY: Apply to the car. EXECUTE: gives.. EVALUATE: The force required is less than the weight of the car by the factor.

Figure 5.1a, b IDENTIFY: Apply to the car. EXECUTE: gives.. EVALUATE: The force required is less than the weight of the car by the factor. 51 IDENTIFY: for each object Apply to each weight and to the pulley SET UP: Take upward The pulley has negligible mass Let be the tension in the rope and let be the tension in the chain EXECUTE: (a) The