( ) ( ) A i ˆj. What is the unit vector Â that points in the direction of A? 1) The vector A is given by = ( 6.0m ) ˆ ( 8.0m ) Solution A D) 6 E) 6

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1 A i ˆj. What is the unit vector Â that points in the direction of A? 1) The vector A is given b ( 6.m ) ˆ ( 8.m ) A ˆ i ˆ ˆ j A ˆ i ˆ ˆ j C) A ˆ ( 1 ) ( i ˆ ˆ j) D) Aˆ.6 iˆ+.8 ˆj E) Aˆ.6 iˆ.8 ˆj A) (.6m ) + (.8m ) B) (.6m ) (.8m ) ( ) ˆ ( ) ˆ ( ) ˆ ( ) ˆ ˆ A 6.m i 8.m j 6.m i 8.m j A.6iˆ.8 ˆj A 6.m + 8.m 1.m ( ) ( ) ) Give ( k ˆ i ˆ ) ( 3 ˆ j) A) 6 î B) 6 ĵ C) 6î D) 6 E) 6 k ˆ i ˆj i. ˆ ( k ˆ ˆ i ) ( 3 ˆj ) ( ˆj ) ( 3 ˆj ) 6 3) In the figure below, what is the average acceleration between and 5. s, and what is the instantaneous acceleration at 5. s? A) and 1 m/s B). m/s and 1 m/s C). m/s and 1 m/s D) 1 m/s and. m/s E) 1 m/s and Δv 1m/s vave.m/s Δt 5s dv m/s m/s v 1m/s dt 8s 4s

2 4) A projectile is shot horizontall at 3.4 m/s from the roof of a 55. m tall building. The projectile hits the level ground at a distance of from the base of the building. A) 55.4 m B) 78.4 m C) 111 m D) 157 m E) 187 m Given: v x 3.4m/s v 55.m x x? For projectile motion: + v t gt / x x + v t Numbers: x ( ) t t ( ) ( ) ( ) t ( ) ( ) ( ) 55.m 4.9m/s 55.m / 4.9m/s x 3.4m/s 3.4m/s 55.m / 4.9m/s 78.4m More formal solution: gt / t / g x x ( ) ( ) ( ) x v t v / g 3.4m/s 55.m / 4.9m/s 78.4m

3 5) A ball is thrown at an original speed of 8. m/s at an angle of 35 above the horizontal. What is the speed of the ball when it returns to the same horizontal level? Neglect air resistance. A) 4. m/s B) 8. m/s C) 9.8 m/s D) 16 m/s E) m/s Given: v 8.m/s v? Method 1 (Kinematics) For projectile motion: + v t gt / v v x v v x gt For this problem: v t gt / t v v v x v v x gt v v g / g ( v / g) v ( ) v vx + v vx + v vx + v v 8.m/s Method (Kinematics) v v g( ) v v v v 8m/s Method 3 (Conservation of energ) Initial energ: E i mg + Final energ: E f mg + Ef Ei mg+ mg + v v 8m/s

4 6) A bullet is fired horizontall, and at the same instant a second bullet is dropped from the same height. Ignore air resistance. Compare the times it takes for the two bullets to hit the ground. A) The fired bullet hits the ground first. B) The dropped bullet hits the ground first. C) The two bullets hit the ground at the same time. D) Cannot tell without knowing the masses of the bullets. E) Cannot tell without knowing the speed of the fired bullet. Horizontal and vertical motions are independent. The two bullets hit the ground at the same time because the vertical component of velocit and the height are the same. 7) An object moves in a circular path at a constant speed. Compare the direction of the object's velocit and acceleration vectors. A) The vectors point in the same direction. B) The vectors point in opposite directions. C) The vectors are perpendicular to each other. D) The question is meaningless, since the acceleration is zero. E) The question is meaningless, because the velocit and acceleration vectors change directions on the circular path. As an object moves in a circular path at a constant speed: Velocit is tangent to the trajector (perpendicular to the radius of the circle). The tangential acceleration is zero because the velocit is constant. The centripetal acceleration is directed to the center (along the radius). Because of that the velocit is perpendicular to the acceleration.

5 8) The angular velocit of a particle that undergoes circular motion as function of time t is given b ω rad/s + ( 3 rad/s ) t. Calculate the angle covered b the particle between t s and t s. A) 8 rad B) 1 rad C) 1 rad D) 16 rad E) rad 1 ( rad/s ( 3rad/s ) ) ( rad/s) ( 3rad/s ) t s s θ ωdt t t dt t t 1 ( rad/s)( s) ( 3rad/s )( s) 1rad/s Δ ) The acceleration of gravit on the Moon is onl one-sixth of that on Earth. If one shoots one cannonball on the Moon and one cannonball on earth both with the same initial speed and launch angle, the ratio of their ranges R moon /R earth would be. A) 1/6 B) 1 C) 6 D) 36 E) None of the above. For projectile motion: + v t gt / x x + v t x To find the time to return to the ground we set. From the first equation follows: + v t gt / t v / g The range is R x x vxt vxv / g 1/ g Finall R / R g / g 6 moon earth earth moon

1) The driver of a motorboat, which can move at a speed of 1 m/s

6 km wide in which the current flows at 5. m/s.

Solving for the right triangle: v boat / shore v boat / water v

6 1 m ) ( 1m/s) / water / shore ( ) 5m/s 185s 3.

elevator. when the elevator is accelerating downward at 4. m/s?

N The force on the person b the scale is the same as the force

6 1) The driver of a motorboat, which can move at a speed of 1 m/s in still water, wishes to travel directl across a river 1.6 km wide in which the current flows at 5. m/s. How long will it take to crosss the river? A) B) C) D) E) 5.3 min 3.1 min.7 min.4 min 1.8 min Solut v tion boat / shore v boat / water + v water / shore Solving for the right triangle: v boat / shore v boat / water v water / shore t d / v boat / shore d v boat water v 3 t (1.6 1 m ) ( 1m/s) / water / shore ( ) 5m/s 185s 3.1min 11) A person of weight 48 N stands on a scale in an elevator. when the elevator is accelerating downward at 4. m/s? What will the scale read A) 196 N B) 84 N C) 48 N D) 676 N E) 74 N The force on the person b the scale is the same as the force on the scale b the person. From the free-bod diagram for the person we get the force N on the person b the scale. 4.m/s mg N ma N mg(1 a/ g ) ( 48N) 1 9.8m/s 84N

7 1) Two horizontal forces act on a 5.-kg mass. One force has a magnitude of 8. N and is directed due north. The second force toward the east has a magnitude of 6.. N. What is the acceleration of the mass? A) 1.6 m/s due north B) 1. m/s due east C). m/s at 53 N of E D). m/s at 53 o E of N E).8 m/s at 45 o N of E ( a F / m 8.N + 6.N ( ) ( ) θ arctan 8N 6.N 53 ) ( 5.kg) ) m/s 13) Two boxes of masses m and 3m are stacked as sketched below. The surface between the more massive box and the horizontal surface is frictionless and the surface between the two boxes is rough. If the less massive box does not slide on the more massive box, what is the static friction force on the less massive box? A) F B) F/ C) F/3 D) F/4 E) Cannot calculate without the coefficient of static friction for the contact surface between the boxes. From the second Newton s law follows: 4ma Fs f ma f F /4. s

8 14) In the Atwood machine shown below, if M. 6 kg and m.4 kg, what is the tension in the string? (Ignore friction and the mass of the pulle.) A).9 N B) 4.7 N C) 6.7 N D) 8.1 N E) 9.8 N Newton s second law for each mass: Mg T Ma + T mg ma Mg mg Ma + ma ( )( ( ) )(.6kg.4kg 9.8m/s T.6kg +.4kg ( ) T mg M m a g M + m ) 4.7N ( 1+ a g) M T mg 1+ M m mmg + m M + m

9 15) An object is placed on an inclined plane. The angle between the incline and the horizontal is graduall increased until the object begins to slide. The angle at which the object begins to slide is θ. What is the coefficient of static friction between the object and the plane? A) 1/ sin θ B) sin θ C) cos θ D) tan θ E) 1/tan θ N f s mg cosθ θ mg sinθ mg θ x a mg sinθ f s N mg cosθ f s mg sinθ N mg cosθ mg sinθ f s μ s N μ f s N tanθ mg cosθ μ tanθ s s max

10 16) A stone, of mass m, is attached to a strong string and whirled in a vertical circle of radius R. At the exact bottom of the path the tension in the string is 3 times the stone's weight. The stone's speed at this point is given b. A) gr B) gr C) gr D) gr E) g/ R T mg Given: T 3mg T mg mg R R v gr

11 17) A red car goes around a curve of radius R at a constant speed v. A blue car of the same make goes around a curve of radius R at speed v. What is the ratio of centripetal forces acting on the two cars F blue / F red when the go around the curves? A) 1/4 B) 1/ C) 1 D) E) 4 F F red blue R ( v) m R F F blue red 18) You keep a book from falling b pushing it against a wall as in the figure below. Which of the following statements is TRUE? A) The normal force on the book b the wall and the normal force on the book b the hand constitute a Newton s 3 rd law force pair. This statement is FALSE. (The pair of Newton s 3 rd law forces should be applied to a pair of interacting objects.) B) The weight of the book and the friction force between the wall and the book constitute a Newton s 3 rd law force pair. This statement is FALSE. (The weight of the book is the force of interaction between the book and the earth.) C) There is no friction force present in this problem because the book does not move. This statement is FALSE. (Without friction the book would slide down the wall.) D) The normal force on the book b the wall and the normal force on the wall b the book constitute a Newton s 3 rd law force pair. This statement is TRUE (This is a pair of forces related to the interaction between two objects.) E) The friction force between the wall and the book and the normal force on the book b the wall constitute a Newton s 3 rd law force pair. This statement is FALSE. (The pair of Newton s 3 rd law forces should have opposite direction.)

12 19) Three objects are in equilibrium in a sstem made of massless ropes andd two ideal frictionless pulles with stationar axles as in the figure below. The mass M of the object in the center is. A) B) C) D) E) M mcosθ M msinθ M mtanθ M m/tanθ M m T l T T r T l T r mg T Mg + T + T cosθ + T T l T r T l T r cosθ T mg cosθ Mg M mcosθ ) A constant angular acceleration of α.5 rad/s is applied too a disk thatt was originall spinning at an angular velocit of rad/s. The disk makes revolutions until it reaches an angular velocit of 1 rad/s. A) 15 B) C) 3 D) 5 E) 1 Given: α.5rad/s ω rad/s ω 1rad/s Δθ π? ω ω αδθ Δθ ω ω 15.3 π ( π )( α )

at a speed of 5 m/s. If both objects have the same kinetic energ, the ratio of the brick's mass to the pebble's mass is m B / m P. A) 5/9 B) 5/3 C) 1.5/4.

13 1) A 1-kg mass is moving with a speed of 5. m/s. the mass? How much work is required to stop A) 5 J B) 75 J C) 1 J D) 15 J E) 5 J Work kinetic energ theorem: W Δ K W ext W ( 1kg)( 5m/s) W W 15JJ ) A brick is moving at a speed of 3 m/s and a pebble is moving at a speed of 5 m/s. If both objects have the same kinetic energ, the ratio of the brick's mass to the pebble's mass is m B / m P. A) 5/9 B) 5/3 C) 1.5/4.5 D) 3/5 E) 5/ K K B K P 1 K B K P m v B m v P B P m m B P v v P B 5 9 3) A car on a level road has a pendulum attached to its ceiling with a string. You observe that the string makes an angle θ with the vertical. Whichh of the following statements is TRUE? A) The car is at rest. B) The car moves to the right of the page. C) The car moves to the left of the page. D) The car accelerates to the right of the page. E) The car accelerates to the left of the page The geometrical sum of two forces (tension and weight) is directed to the left. This means that the car accelerates to the left of the page.

14 4) Two blocks of masses m and 4m are connected through a massless, ideal string that goes through two ideal, massless pulles in the arrangement below. There is no friction between the blocks and the surfaces. The tension in the string is. A) 4/ /5 mg B) 8/ /5 mg C) 3 mg D) 4 mg E) 5 mg T Free bod diagram: Second Newton s law: mg 4mg 4mg T 4ma + T mg ma 3mg 5 ma T m(a + 3 a 5 g g) T 8 / 5mg 5) Which of the free-bod diagrams below represents a block sliding down a frictionless inclined surface? Answer D represents a block sliding down a frictionless inclined surface. Weight is directed down, and the normal force is directed perpendicular to the inclined surface.

15 6) Which of the following statements is FALSE? (In the following we assume that the internal energ of the object remains unchanged.) A) If nonzero net work was done on an object, then its speed must have changed. B) If nonzero net work was done on an object, then its velocit must have changed. C) If no net work was done on an object, then its direction of motion has not changed. This statement is FALSE. We have a counter example: if a force is perpendicular to the displacement then work done b this force is equal to zero, but this force is changing the direction of the motion of the object. No work is done on an object in uniform circular motion (since its speed does not change), but it changes direction all the time. D) No work is done b the forces acting on an object if it remains at rest. E) A force that is perpendicular to the velocit of an object doesn t do work on the object. 7) A marble with mass 1 gram is fired from a spring loaded to cannon with a speed of. m/s. The spring constant of the cannon spring is 1 N/m. How far was the spring compressed before the shot? A). cm B) 4. cm C) 16 cm D) cm E) 4 cm Given: m 1g v.m / s k 1N / m x? Δ K kx W W ΔK kx x v m k x kg. 1 m x.cm (.m/s) 1N/m

PHYSICS 221, FALL 2009 EXAM #1 SOLUTIONS WEDNESDAY, SEPTEMBER 30, 2009

PHYSICS 221, FALL 2009 EXAM #1 SOLUTIONS WEDNESDAY, SEPTEMBER 30, 2009 Note: The unit vectors in the +x, +y, and +z directions of a right-handed Cartesian coordinate system are î, ĵ, and ˆk, respectively.