AP Physics C Mechanics Free Response 2017

2. A block of mass m starts at rest at the top of an inclined plane of height h, as shown in the figure above. The block travels down the inclined plane and makes a smooth transition onto a horizontal surface. While traveling on the horizontal surface, the block collides with and attaches to an ideal spring of spring constant k. There is negligible friction between the block and both the inclined plane and the horizontal surface, and the spring has negligible mass. Express all algebraic answers for Parts A, B, and C in terms of m, h, k, and physical constants, as appropriate. (a) (i) Derive an expression for the speed of the block just before it collides with the spring. (ii) Is the speed halfway down the incline greater than, less than, or equal to one-half the speed at the bottom of the inclined plane? Justify your answer.

(a) (i) For correctly using conservation of energy for the block moving down the incline *Conservation of Mechanical Energy* *no external, all conservative internal forces* U=K mgh=½mv² For a correct answer *v=√(2gh)* (ii) Correct answer: "*Greater than*" For a correct justification *Conservation of Mechanical Energy* conditions still apply U_(half)+K=U mg(½h)+½mv²=mgh v_(half)=√(gh) v=√2[v_(half)] *v_(half)>*½v=½√2[v_(half)]=*0.705[v_(half)]* Example: The speed is proportional to the square root of the change in height. So if the height is reduced by a factor of 2, the speed is reduced by a factor of √2≈1.41 Therefore, the speed halfway down the ramp is more than half the speed at the bottom of the ramp. Note: If the incorrect selection is made, the justification cannot earn credit.

3. A uniform solid cylinder of mass M = 0.50 kg and radius R = 0.10 m is released from rest, rolls without slipping down a 1.0 m long inclined plane, and is launched horizontally from a horizontal table of height 0.75 m. The inclined plane makes an angle of 30° with the horizontal. The cylinder lands on the floor a distance D away from the edge of the table, as shown in the figure above. There is a smooth transition from the inclined plane to the horizontal table, and the motion occurs with no frictional energy losses. The rotational inertia of a cylinder around its center is MR²/2. (a) Calculate the total kinetic energy of the cylinder as it reaches the horizontal table.

(a) For correctly applying conservation of energy to the cylinder rolling down the incline *Conservation of Mechanical Energy* *no external, all conservative internal forces* U_g,top=K_table mgh = K_table = (0.50kg)(9.8 m/s²)(1.0m)(sin30) For a correct answer with units K_table= 2.45 J (or 2.5 J using g=10m/s²) h = (1.0m)(sin30)

3. A uniform solid cylinder of mass M = 0.50 kg and radius R = 0.10 m is released from rest, rolls without slipping down a 1.0 m long inclined plane, and is launched horizontally from a horizontal table of height 0.75 m. The inclined plane makes an angle of 30° with the horizontal. The cylinder lands on the floor a distance D away from the edge of the table, as shown in the figure above. There is a smooth transition from the inclined plane to the horizontal table, and the motion occurs with no frictional energy losses. The rotational inertia of a cylinder around its center is MR²/2. (b) Calculate the angular velocity of the cylinder around its axis at the moment it reaches the floor.

(b) For correctly setting the kinetic energy of the cylinder equal to the sum of both the linear and rotational kinetic energy *K=½mv² + ½Iω²* *v=rω* For correctly substituting into the above equation for the linear velocity and moment of inertia of the cylinder K=½MR²ω² + ½(½MR²)ω² =½M(Rω)² + ¼M(Rω)² =(3/4)M(Rω)² so ω=√([4k]/[3MR²]) =√([4(2.45J)]/[3(0.50kg)(0.10m)²]) For correct substitution into the equation above ω = 25.6 rad/s (or 26.0 rad/s using g=10 m/s²)

2. A block of mass m starts at rest at the top of an inclined plane of height h, as shown in the figure above. The block travels down the inclined plane and makes a smooth transition onto a horizontal surface. While traveling on the horizontal surface, the block collides with and attaches to an ideal spring of spring constant k. There is negligible friction between the block and both the inclined plane and the horizontal surface, and the spring has negligible mass. Express all algebraic answers for Parts A, B, and C in terms of m, h, k, and physical constants, as appropriate. (b) Derive an expression for the maximum compression of the spring.

(b) For correctly using conservation of energy, consistent with Part A, for the block compressing the spring *Conservation of Mechanical Energy* conditions still apply U_g=U_s *mgh=½kx²* x=√([2mgh]/[k])

2. A block of mass m starts at rest at the top of an inclined plane of height h, as shown in the figure above. The block travels down the inclined plane and makes a smooth transition onto a horizontal surface. While traveling on the horizontal surface, the block collides with and attaches to an ideal spring of spring constant k. There is negligible friction between the block and both the inclined plane and the horizontal surface, and the spring has negligible mass. Express all algebraic answers for Parts A, B, and C in terms of m, h, k, and physical constants, as appropriate. (c) Determine an expression for the time from when the block collides with the spring to when the spring reaches its maximum compression.

(c) For indicating a *simple harmonic motion* approach For a correct answer t=¼T=¼[2π√(m/k)]=(π/2)√(m/k) *T=2π√(m/k)* (where T is period)

1. An Atwood's machine consists of two blocks connected by a light string that passes over a frictionless pulley of negligible mass, as shown in the figure above. The masses of the two blocks, M₁ and M₂ , can be varied. M₂ is always greater than M₁. The magnitude of the acceleration a was measured for different values of M₁ and M₂, and the data are shown below. M1 (kg) | M2 (kg) | a (m/s2) 1.0 | 2.0 | 3.02 2.0 | 3.0 | 1.82 5.0 | 12.0 | 4.21 6.0 | 8.0 | 1.15 10.0 | 14.0 | 1.71 (c) Indicate below which quantities should be graphed to yield a straight line whose slope could be used to calculate a numerical value for the acceleration due to gravity g. Vertical axis: _____ Horizontal axis: _____ Use the remaining rows in the table above, as needed, to record any quantities that you indicated that are not given.

(c) For indicating variables that will create a straight line whose slope can be used to determine g Example: Vertical (Y) axis: a Horizontal (X) axis: [(M₂-M₁)/(M₁+M₂)] *found in Part B a=[(M₂-M₁)/(M₁+M₂)]g* Note: Full credit is earned if axes are reversed, or if the student uses another acceptable combination.

3. A uniform solid cylinder of mass M = 0.50 kg and radius R = 0.10 m is released from rest, rolls without slipping down a 1.0 m long inclined plane, and is launched horizontally from a horizontal table of height 0.75 m. The inclined plane makes an angle of 30° with the horizontal. The cylinder lands on the floor a distance D away from the edge of the table, as shown in the figure above. There is a smooth transition from the inclined plane to the horizontal table, and the motion occurs with no frictional energy losses. The rotational inertia of a cylinder around its center is MR²/2. (c) Calculate the ratio of the rotational kinetic energy to the total kinetic energy for the cylinder at the moment it reaches the floor.

(c) For using a correct expression for the ratio of the rotational kinetic energy to the total kinetic energy of the cylinder *K_tot= Kinetic + Rotation + Potential* K_rot/K_tot =[½Iω²] / [½mv² + ½Iω² + Mgh] = [¼M(Rω)²] / [(3/4)M(Rω)² + Mgh] = [(Rω)²] / [3(Rω)² + 4gh] For substituting into the above equation [([0.10m][25.6rad/s])²] / [(3)(0.10m)²(25.6rad/s)² + (4)(9.81m/s²)(0.75m)] K_rot/K_tot=0.133 (or 0.135 using g=10 m/s²)

2. A block of mass m starts at rest at the top of an inclined plane of height h, as shown in the figure above. The block travels down the inclined plane and makes a smooth transition onto a horizontal surface. While traveling on the horizontal surface, the block collides with and attaches to an ideal spring of spring constant k. There is negligible friction between the block and both the inclined plane and the horizontal surface, and the spring has negligible mass. Express all algebraic answers for Parts A, B, and C in terms of m, h, k, and physical constants, as appropriate. The block is again released from rest at the top of the incline, and when it reaches the horizontal surface it is moving with speed v₀ . Now suppose the block experiences a resistive force as it slides on the horizontal surface. The magnitude of the resistive force ƒ is given as a function of speed v by ƒ = βv² , where β is a positive constant with units of kg/m . (d) (i) Write, but do NOT solve, a differential equation for the speed of the block on the horizontal surface as a function of time t before it reaches the spring. Express your answer in terms of m, h, k, β, v, and physical constants, as appropriate. (ii) Using the differential equation from Part D₁, show that the speed of the block v(t) as a function of time t can be written in the form 1/v(t) = 1/v₀ + βt/m where v₀ is the speed at t=0

(d) (i) For correctly applying Newton's second law for the horizontally sliding block. *F_net=ma* For correctly indicating that the direction of *friction is opposite the direction* of motion. given F_net =ƒ= βv² -βv²=ma For expressing the equation as a differential equation *a=(dv/dt)* -βv²=m(dv/dt) (ii) *Separable Differentiable Equations* For correctly separating variables -β/m dt = 1/v² dv For correctly integrating the equation above ∫-β/m dt = ∫1/v² dv (-β/m)(t)=(-1/v) *+ C* For using the correct limits or constant of integration (-β/m)(t)=(-1/v) + C given v₀ t=0 0=(-1/v₀) + C C = 1/v₀ (-β/m)(t)=(-1/v) + 1/v₀ 1/v(t) = 1/v₀ + βt/m

3. A uniform solid cylinder of mass M = 0.50 kg and radius R = 0.10 m is released from rest, rolls without slipping down a 1.0 m long inclined plane, and is launched horizontally from a horizontal table of height 0.75 m. The inclined plane makes an angle of 30° with the horizontal. The cylinder lands on the floor a distance D away from the edge of the table, as shown in the figure above. There is a smooth transition from the inclined plane to the horizontal table, and the motion occurs with no frictional energy losses. The rotational inertia of a cylinder around its center is MR²/2. (d) Calculate the horizontal distance D.

(d) For correctly using motion in the vertical direction to calculate the time for the cylinder to reach the floor *y = y₀ + v₀t + ½at²* y-y₀ = 0 - ½gt² Determine the time for the cylinder to reach the floor y-y₀ =- ½gt² *let y=floor=0* *let y₀=table height=yₙ=0.75m* 0-yₙ =- ½gt² solve for t t=√(2yₙ/g)=√(2(0.75m))/9.81m/s²) = 0.39s For correctly using the equation for constant speed in the horizontal direction *x = x₀ + v₀t + ½at²* x-x₀=v₀t *v=rω* x-x₀=rωt x-x₀=rω√(2yₙ/g) D=rωt=(0.10m)(25.6rad/s)(0.39s)=1.0m

2. A block of mass m starts at rest at the top of an inclined plane of height h, as shown in the figure above. The block travels down the inclined plane and makes a smooth transition onto a horizontal surface. While traveling on the horizontal surface, the block collides with and attaches to an ideal spring of spring constant k. There is negligible friction between the block and both the inclined plane and the horizontal surface, and the spring has negligible mass. Express all algebraic answers for Parts A, B, and C in terms of m, h, k, and physical constants, as appropriate. (e) Sketch graphs of position [x] as a function of time [t], velocity [v] as a function of time [t], and acceleration [a] as a function of time [t] for the block as it is moving on the horizontal surface before it reaches the spring.

(e) (i) For a *displacement curve* that is *concave down, increasing* and approaching a *horizontal asymptote* *In Q1* (ii) For a *velocity curve* that is *concave up, decreasing* and has the *horizontal axis as an asymptote* *In Q1* (iii) For an *acceleration curve* that is *concave down, increasing* and has the *horizontal axis as an asymptote* *In Q2* Note: If an incorrect nonlinear velocity graph is generated, 1 point is earned if the position and acceleration graphs are consistent with the velocity graph. Note: Full credit is earned if all three graphs are flipped vertically.

3. A uniform solid cylinder of mass M = 0.50 kg and radius R = 0.10 m is released from rest, rolls without slipping down a 1.0 m long inclined plane, and is launched horizontally from a horizontal table of height 0.75 m. The inclined plane makes an angle of 30° with the horizontal. The cylinder lands on the floor a distance D away from the edge of the table, as shown in the figure above. There is a smooth transition from the inclined plane to the horizontal table, and the motion occurs with no frictional energy losses. The rotational inertia of a cylinder around its center is MR²/2. A sphere of the same mass and radius is now rolled down the same inclined plane. The rotational inertia of a sphere around its center is (2/5)MR². (e) (i) Is the total kinetic energy of the sphere at the moment it reaches the floor greater than, less than, or equal to the total kinetic energy of the cylinder at the moment it reaches the floor? Justify your answer. (ii) Is the rotational kinetic energy of the sphere at the moment it reaches the floor greater than, less than, or equal to the rotational kinetic energy of the cylinder at the moment it reaches the floor? Justify your answer. (iii) Is the horizontal distance the sphere travels from the table to where it hits the floor greater than, less than, or equal to the horizontal distance the cylinder travels from the table to where it hits the floor? Justify your answer.

(e) (i) For selecting *"Equal to"* and attempting a relevant justification For a correct justification Example: Because the sphere falls the same height as the cylinder and because they have the same mass, the sphere-Earth system has the same initial potential energy and, therefore, the *same total kinetic energy* when it reaches the floor. *m₁=m₂* *m₁gh=m₂gh* (ii) For selecting *"Less than"* and attempting a relevant justification For a correct justification Example: Because the rotational inertia of the sphere is less than the rotational inertia of the cylinder, the sphere will rotate faster and, because v=rω , will move with a greater linear speed. Because the mass is the same and the linear speed is greater, the sphere will have a greater linear kinetic energy. Because the total kinetic energies of the sphere and cylinder are the same, the *sphere must have less rotational kinetic energy*. *Less rotational since I_s>I_c* K_s=(1/2)(2/5)M(rω)²=(1/5)M(rω)² K_c=(1/2)(1/2)M(rω)²=(1/4)M(rω)² K_s=(5/4)(K_c) so K_s has a greater linear kinetic energy (speed) but found total kinetic energies are the same so sphere must have less rotational kinetic energy (iii) For selecting *"Greater than"* and attempting a relevant justification For a correct justification Example: Because the *sphere has a greater linear speed* as it leaves the table, it will travel a greater horizontal distance before it reaches the floor. *sphere rotates less (Part B) so it has more linear K*

1. An Atwood's machine consists of two blocks connected by a light string that passes over a frictionless pulley of negligible mass, as shown in the figure above. The masses of the two blocks, M₁ and M₂ , can be varied. M₂ is always greater than M₁. The magnitude of the acceleration a was measured for different values of M₁ and M₂, and the data are shown below. M1 (kg) | M2 (kg) | a (m/s2) 1.0 | 2.0 | 3.02 2.0 | 3.0 | 1.82 5.0 | 12.0 | 4.21 6.0 | 8.0 | 1.15 10.0 | 14.0 | 1.71 (e) Using your straight line, determine an experimental value for g.

(e) For correctly calculating the slope *using the best-fit line* and not the data points, unless the points fall on the best-fit line (Note: Credit may be given for the linear regression only if the student states linear regression is used.) slope=(y₂-y₁)/(x₂-x₁) =[(3.0-1.0)m/s²]/[0.31-0.12] =10.5 m/s² For correctly relating the slope to g g = slope = 10.5 m/s² (using linear regression yields 10.1 m/s²)

1. An Atwood's machine consists of two blocks connected by a light string that passes over a frictionless pulley of negligible mass, as shown in the figure above. The masses of the two blocks, M₁ and M₂ , can be varied. M₂ is always greater than M₁. The experiment is now repeated with a modification. The Atwood's machine is now set up so that the block of mass M₁ is on a smooth, horizontal table and the block of mass M₂ is hanging over the side of the table, as shown in the figure above. (f) For the same values of M₁ and M₂, is the magnitude of the tension in the string when the blocks are moving higher, lower, or equal to the magnitude of the tension in the string when the blocks are moving in the first experiment?

(f) Correct answer: *"Lower"* For including a correct statement about the acceleration of the blocks. For including a correct statement about the forces on the blocks Example: Because the block is on the table, the net force on the system increases, and therefore the acceleration increases. Because the acceleration of increases, the net force on must increase. Therefore, there must be a greater difference in the magnitude of the two forces on. Because the weight of block stays the same, the retarding force — the tension in the string — must decrease. *M₁ no longer pulling down* T-mg=-ma so *T=m(g-a)*

1. An Atwood's machine consists of two blocks connected by a light string that passes over a frictionless pulley of negligible mass, as shown in the figure above. The masses of the two blocks, M₁ and M₂ , can be varied. M₂ is always greater than M₁. The experiment is now repeated with a modification. The Atwood's machine is now set up so that the block of mass M₁ is on a smooth, horizontal table and the block of mass M₂ is hanging over the side of the table, as shown in the figure above. (g) The value determined for the acceleration due to gravity g is lower than in the first experiment. Give one physical factor that could account for this lower value and explain how this factor affected the experiment.

(g) For a correct justification *What is interfering?* Example: Block will experience *friction* with the table. The acceleration of the system will decrease and this will decrease the slope of the line; therefore, the value of g is determined by the experiment.

1. An Atwood's machine consists of two blocks connected by a light string that passes over a frictionless pulley of negligible mass, as shown in the figure above. The masses of the two blocks, M₁ and M₂ , can be varied. M₂ is always greater than M₁. (b) Using the forces in your diagrams above, write an equation applying Newton's second law to each block and use these two equations to derive the magnitude of the acceleration of the blocks and show that it is given by a=[(M₂-M₁)/(M₁+M₂)]g

(i) For correctly applying Newton's second law to block 1. T-M₁g=M₁a (ii) For correctly applying Newton's second law to block 2 M₂g-T=M₂a *∑F=Ma* (iii) For combining the two equations above in such a way that will lead to the correct equation M₂g-M₁g=(M₁+M₂)a a=[(M₂-M₁)/(M₁+M₂)]g

1. An Atwood's machine consists of two blocks connected by a light string that passes over a frictionless pulley of negligible mass, as shown in the figure above. The masses of the two blocks, M₁ and M₂ , can be varied. M₂ is always greater than M₁. (a) On the dots below, which represent the blocks, draw and label the forces (not components) that act on the blocks. Each force must be represented by a distinct arrow starting on and pointing away from the appropriate dot. The relative lengths of the arrows should show the relative magnitudes of the forces.

(i) For correctly drawing and labeling the vectors for the *weight* of the block and the *tension* for block M₁ with the tension larger than the weight (ii) For correctly drawing and labeling the vectors for the weight of the block and the tension for block M₂ with the weight larger than the tension (iii) For correctly drawing tension on the two blocks as equal in magnitude *weight₁<T₁=T₂<weight₂* Note: If any extraneous vectors are drawn, only a maximum of two points may be earned.

1. An Atwood's machine consists of two blocks connected by a light string that passes over a frictionless pulley of negligible mass, as shown in the figure above. The masses of the two blocks, M₁ and M₂ , can be varied. M₂ is always greater than M₁. The magnitude of the acceleration a was measured for different values of M₁ and M₂, and the data are shown below. M1 (kg) | M2 (kg) | a (m/s2) 1.0 | 2.0 | 3.02 2.0 | 3.0 | 1.82 5.0 | 12.0 | 4.21 6.0 | 8.0 | 1.15 10.0 | 14.0 | 1.71 (d) Plot the data points for the quantities indicated in Part C on the graph below. Clearly scale and label all axes including units, if appropriate. Draw a straight line that best represents the data.

For using a *correct scale* that uses more than half the grid and for correctly *label*ing the axes (*0→5,0→0.5*), including *units* as appropriate For correctly plotting the data For drawing a straight best-fit line consistent with the plotted data