# Solution Only gravity is doing work. Since gravity is a conservative force mechanical energy is conserved:

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1 8) roller coaster starts with a speed of 8.0 m/s at a point 45 m above the bottom of a dip (see figure). Neglecting friction, what will be the speed of the roller coaster at the top of the next slope, which is 0 m above the bottom of the dip? ) 8 m/s ) m/s C) 7 m/s D) 9 m/s E) m/s Solution Only gravity is doing work. Since gravity is a conservative force mechanical energy is conserved: E E mv mgh mv + mgh v v + g ( h h ) i f i + i f f f i i f ( 8.0m / s) + ( 9.8m / s )( 45m ) v f 0m v f 9m / s 9) block with mass m.0 kg is released from rest at the top of an incline. The incline makes an angle of 0 with the horizontal. The coefficient of kinetic friction between the block and the incline is 0.0. What will be the speed of the block after sliding 4.0 m along the incline? ). m/s ).0 m/s C).5 m/s D) 5. m/s E) 6.0 m/s h d sinθ N mg cosθ f μ N μ mg cosθ k k k

2 Method (using Newton s Second Law): a mg sinθ f m g sinθ μ cosθ v ( ) ( ) v f i k ad v k v + ad v gd( sinθ μ cosθ ) f i ( 9.8m / s )( 4.0 )( sin 0 0.0cos 0 ) v f m v f.5m / s f k Method (using conservation of energy): E E W mv mgh f d f i friction k mv mgd θ μ mg θd sin k cos ( ) v gd sinθ μ cosθ.5m/s f k f k mgcosθ θ N mgsinθ mg θ 0).00-kg block is on a frictionless surface and originally at rest. 0.0-g bullet moving at 00 m/s is fired into the block. The bullet emerges on the opposite side of the block with half the bullet s original speed and the direction of the original velocity. What is the speed of the block after the collision? ).50 m/s ).97 m/s C) 5.0 m/s D) 7 m/s E) 00 m/s Conservation of linear momentum: m mv Mvb + m( v) v b v M v b kg ( 00 m/ s).00 kg vb.50 m/ s

3 ) The graph below shows the potential energy of a -kg particle confined to move along the x-axis as a function of position. The particle is moving to the right at point O with a kinetic energy K 0J. Which of the labeled points is a turning point for the motion of the particle? ) Point ) Point C) Point C D) Point D E) Point E t point O, the total mechanical energy is E UO + KEO 5J + 0J 5J. t a turning point the kinetic energy is zero and the potential energy is equal to the total mechanical energy. t point E: U E 5J KE E U 0 Point E is a turning point. E E ) The cross section of a piece of metal of uniform density is shown in the figure below. The y-position of the center of mass of the piece is at. ) 0.7 m ) 0.8 m C) 0.9 m D).0 m E). m Let M be the mass of one square of the metal piece and let y n be the y coordinate of the center of n s square, then Myn yn 5 ycm m.m M 0 0

4 ) If a solid cylinder rolls without slipping, the ratio of its rotational kinetic energy to its translational kinetic energy is E rot / E trans. ) : ) : C) : D) : E) : For a solid cylinder: I mr For rolling without slipping: ω v / R E E rot trans Iω mv 4 mv / E rot E trans 4) Two blocks slide on a frictionless surface in the positive x direction. lock has mass m and block has mass m m. efore the two blocks collide, block has velocity v 6 m/s block has velocity v m/s. fter the collision, which is elastic and one-dimensional, the velocity of block is. Α) 4 m/s ) m/s v v C) 0 D) + m/s E) +4 m/s m m Conservation of linear momentum for collision: m v + mv mv' + mv' m(6m/s) + m(m/s) mv ' + mv ' v' + v' m/s () For elastic collision: v v ( v' v ) ' ( v v ) ' ' 4m/s () Solving equations () and () we have: v ' 0 and v ' 4m/s.

5 5) red car of mass m r 500 kg is moving with a velocity of v r 90.0 km/h due east and a blue car of mass m b 000 kg is moving with a velocity of v b 60.0 km/h due south. The two cars collide and stick together after the collision. What is the speed of the two car system immediately after the collision? ) 0 km/h ) 50 km/h C) 6 km/h D) 78 km/h E) 0 km/h This is a completely inelastic collision. Linear momentum is conserved in the collision as no net external force acts on the two car system. mrvr + mbvb r b v ecause v r v, ( m + m ) b mrv v m r r + mbv + m iˆ ˆj v km/h ( 0iˆ+ 40 ˆj) km/h ( ( ) ( ) ) v km/h 50km/h b b or v ( mv ) + ( mv ) r r b b m r + m b ( 500kg)( 90.0km/h) + ( 000kg)( 60.0km/h) v 50km/h 500kg + 000kg

6 6) Four small balls are connected in form of a rectangle by massless rods. The long sides of the rectangle are twice as long as the short sides. The rectangle can rotate around several axes as shown below. Which rotation axis yields the smallest moment of inertia? ) xis through balls on long side ) xis through centers of short sides C) Perpendicular to plane of rectangle, through center of rectangle D) xis through balls on short side E) xis through centers of long sides Let the length of the short side be r, long side r, and mass of each ball m, then I mr I I I I C D E 4m r 4m m 4m r ( ) mr ( + 4 ) r 5 ( r) 8mr () 4mr mr 7).5-kg mass hangs on a rope wrapped around a disk pulley of mass 7.07 kg and radius 66.0 cm. The rope does not slip on the pulley. What is the angular acceleration of the pulley? ) 4.49 rad/s ) 7.98 rad/s C) 9.87 rad/s D). rad/s E) zero M7.07 T mg T ma mg mα R + T TR Iα T MαR ( ) ( m + M ) α mg R α + m.5 kg g M ( m)r 9.80m/s α α 4.49rad/s m mg

7 8) uniform meter stick of mass 0. kg is supported by a knife edge at the 50-cm mark and has masses of 0.40 kg and 0.60 kg hanging at the 0-cm and 80-cm marks, respectively. third mass of 0.0 kg is hung at the mark to keep the stick in balance. ) 0-cm ) 5-cm C) 0-cm D) 70-cm E) 80-cm We consider rotation around the point at the 50-cm mark. For balance (static equilibrium) the net torque around this point has to be zero: ( 0.40 )( 50cm 0cm) + ( 0.60kg)( 50cm 80cm) + ( 0.0kg)( 50cm x) 0 kg x 0cm 9) n aluminum wire.0 m in length and.0 mm in diameter hangs from the ceiling. How much does the wire stretch when a0.0-kg mass is hung off it? (The Young's modulus for aluminum is N/m.) ) 0. mm ) 0. mm C) 0. mm D) 0.89 mm E) 0.99 mm y definition of the Young's modulus: Y F / ΔL / L Δ L LF Y Lmg Yπr ΔL (.0m)( 0.0kg)( 9.8m/ s ) 0 ( N / m ) π (.0 0 m) ; ΔL m 0. 89mm.

8 The situation described below pertains to the next two questions: uniform rod of mass.0 kg and length 4.00 m is attached at one end with a hinge to a wall. The other end is supported by a massless cable attached to the ceiling. The angle between the ceiling and the cable is 60 o (see figure). 40) What is the tension in the cable? ) 68 N ) 6 N C) 49 N D) 8 N E) 04 N T mg We consider rotation about the hinge. The static equilibrium condition for the torque: ( 60 ) mgl / TL sin 0 (.0kg)( 9.8m / s ) T ; T 68N mg mg T sin ( 60 ) ; 4) The cable suddenly snaps. What is the angular acceleration of the rod as it is released? ).68 rad/s ) 4.90 rad/s C) 6.75 rad/s D) 9.4 rad/s E) 4.7 rad/s Moment of inertia for the rod with respect to the end point: I I + m L ml + ml ml CM ( ) 4 Second Newton s law for rotation: τ I α α τ / I mgl / α ml α g L 9.8m / s α ; 4m α.68rad/s

9 4) Consider a hollow sphere where the mass M is uniformly concentrated on the surface. If the sphere is of radius R, what is the moment of inertia around an axis that is tangent to the surface of the sphere? ) (/) MR² ) MR² C) (7/5) MR² D) (5/) MR² E) MR² I MR ; cm I I MR cm + I 5 MR 4) ball of putty with mass m 0.4 kg and speed v 5.0 m/s hits a rod of mass M 0.6 kg and length L 4.0 m that is suspended from the ceiling by a hinge as shown in the figure. The ball hits the rod half-way between the hinge and its bottom end and sticks to the rod after the collision. What is the angular velocity of the ball-rod system after the collision? ) 0. rad/s ) 0.8 rad/s C).4 rad/s D).7 rad/s E). rad/s I I + ML I ( L / ) ML ( / ) + ML ( m M ) L rod CM ML + M ball + I rod m L 4 + Use conservation of angular momentum mvl L i L f mvl / Iω ω I ( 0.4kg)( 5.0m / s) ( 0.4kg + 0.6kg)( 4.0m) ω ; 4 ω mv ( m + M )L 5 ω rad/s 0.8rad/s 6 4

10 44) -kg gold bar falls off a pirate ship and sinks to the bottom of the ocean, where the pressure is 500 atm. The volume of the gold bar at the bottom of the ocean is than it was on the pirate ship. (The bulk modulus of gold is 80 GPa.) a % smaller b. 0.08% smaller c. 0.8% smaller d. the same e. 0.8% larger ΔP ΔV ΔP ΔV / V V Δ V % V ; 5 ( 500 ).0 0 Δ V 9 V 80 0 Pa Pa ; 45) solid cylinder with moment of inertia with respect to its center of mass of I 5 kg m and mass m 0 kg is pivoted about an axle through its center, O. rope wrapped around the outer radius R.0 m, exerts a force F 5.0 N to the right. second rope wrapped around another section of radius R 0.50 m exerts a force F 6.0 N downward. How many radians does the cylinder rotate through in the first 5.0 seconds, if it starts from rest? ).5 rad ) 5.0 rad C) 7.5 rad D) rad E) 0 rad Second Newton s law for rotation: τ I α τ / I θ αt α ( 5.0N.0m 6.0N 0.5m)( 5.0s) (5 kg m ) FR FR α. I α θ ( FR FR ) t. I ; α 5.0rad.

11 46) Planet X has one moon which moves in a circular orbit around the planet with an orbit radius of 4 0 km and a period of.77 days. What is the mass of Planet X? ). 0 7 kg ) kg C) kg D) kg E) kg π R 4π R T M GM TG M 4π 8 ( 4. 0 m) ( s) Nm / kg ; 7 M.9 0 kg 47) Two planets have the same surface gravity (the same g), but planet has twice the mass of planet. If planet has radius R, what is the radius of planet? ) R / ) R / C) R D) R E) R M GM R M g G R R g R M R R

12 48) horizontal disk (Disk ) of radius R and mass M rotates with an angular speed ω 0. nother disk (Disk ) with the same rotation axis, same radius, mass M ¼ M and no initial angular velocity is dropped on top of the first one. fter the drop Disk sticks to Disk and they rotate with a common angular speed of ω. ) zero ) / ω 0 C) / ω 0 D) /4 ω 0 E) 4/5 ω 0 ω 0 Disk Disk I disk mr Use conservation of the angular momentum. I M L i L f I ω0 ( I + I ) ω ω ω0 ω0 I + I M + M ω 4/ 5ω 0 49) plane, flying horizontally, releases a bomb, which explodes before hitting the ground. Neglecting air resistance, the center of mass of the bomb fragments, just after the explosion ) does not move. ) moves horizontally. C) moves vertically. D) moves along a parabolic path. E) none of the above Fext macm the motion of the center of mass does not change after explosions and collisions. The center of mass of the bomb continues to fall freely and move along a parabolic path after the explosion.

13 50) 500-kg car moving at 5 m/s hits an initially uncompressed horizontal spring with spring constant of N/m. What is the maximum compression of the spring? (Neglect the mass of the spring.) ) 0.7 m ) 0.4 m C) 0.5 m D) 0.68 m E) 0.80 m Use conservation of mechanical energy: E E mv kx x v m k ; i f ( ) 6 x 5 m/ s 500 kg (.0 0 N/ m) x 0. 68m.

14 The situation described below pertains to the next two questions: Several forces F are applied to the rods, and C below. ll forces have equal magnitude. ll rods have the same length and uniformly distributed mass. [] [] [C] 5) Rank the net torque about the center of mass for each of the rods. ) τ > τ > τc ) τ C > τ > τ C) τ > τc > τ D) τ > τ > τc E) τ C τ > τ Let r be the length the rod. τ τ rf( + sinθ ) rf τ C 0 5) Rank the net force for each of the rods. ) F > F > FC ) FC > F > F C) F > F FC D) F > FC > F E) FC F > F F F F F ( θ / ) ( / ) F cos 45 0 < θ < 90 is angle between two forces F cos, where F cos θ F. θ θ C C < C F > F C

15 5) Which statement about types of energies, forces and work is TRUE? ) The kinetic energy of a rigid body depends only on its center of mass speed. ) Potential energy is always positive. C) Conservative forces do only positive work. D) Static friction can only do negative work. E) Total mechanical energy of an isolated system is conserved if only conservative forces do work. ) is false: a rigid body can have rotational kinetic energy even when its center of mass is at rest. ) is false. The absolute value of potential energy is a matter of convention. It can be negative. Physics only depends on changes in potential energy. C) is false. For example, gravity decelerates an object that is thrown vertically up. The force is anti-parallel to the displacement. Thus, gravity is doing negative work here. D) is false. Imagine a block on a truck. When the truck accelerates, so does the block. The force that accelerates the block is the static friction between the truck and the block. For this static friction, the force goes in the direction of the displacement and the work done on the block by the static friction is positive. E) is true. 54) diver jumps off a board into a pool rotating around his center of mass. t point (see figure) the diver can be described roughly as a disk with radius R and uniform mass distribution. t point 7 the diver can be described roughly as a rod of length L R and uniform mass distribution. t point his angular speed is ω. What is his angular speed at point 7? ) ω/ ) / ω C) /4 ω D) 9/6 ω E) ω I disk mr ( R) I rod ml m 4 mr Use conservation of angular momentum: L L I ω ω ω ( I I ) ω ω i f disk I rod f f disk rod ω f

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