Newton’s Second Law AP Physics C Mechanics FRQ – Exam Style Questions etc.
Newton’s SECOND Law AP Physics C Mechanics FRQ
Unit 2: Force and Translational Dynamics
Weightage : 20-15%
Question
A student drops a sphere of mass m from rest. The air exerts a drag force of magnitude \(F_{drag}\) on the sphere, as shown in Figure 1. The student models the magnitude of the drag force as \(F_{drag}\) = bv, where v is the speed of the sphere and b is a positive constant with appropriate units.
a) Derive, but do NOT solve, a differential equation that could be used to determine the speed v of the sphere as a function of time t. Express your answer in terms of given quantities and physical constants, as appropriate.
(b) The student sketches the drag force \(F_{drag}\) exerted on the sphere as a function of time t, as shown in Figure 2.
i. Draw a vertical line on the sketch in Figure 2 to indicate the earliest time at which \(F_{drag}\) is equal to the magnitude of the weight of the sphere, which occurs when the sphere reaches terminal speed. Label this time as \(t_{T}\) on the time axis.
ii. Justify the location of \(t_{T}\) . Explicitly reference appropriate features of the sketch in Figure 2.
(c) Suppose the student throws the same sphere downward with a nonzero initial speed. The magnitude of the new drag force at terminal speed after being thrown downward is \(F_{new}\).
Indicate whether \(F_{new}\) would be greater than, less than, or equal to the magnitude of \(F_{drag}\) at terminal speed represented in Figure 2.
_____ Greater than _____ Less than _____ Equal to
Briefly justify your answer.
(d) The student conducts an experiment to better understand the relationship between \(F_{drag}\) and v. The student makes measurements to calculate and graph the magnitude of \(F_{drag}\) as a function of v for the falling sphere.
i. Draw the best-fit line for the data.
ii. Use the best-fit line to calculate an experimental value for b.
A student claims that the terminal speed \(v_{T}\) of the sphere depends on the diameter D of the sphere. The student designs an experiment to collect data that can be used to provide evidence to support the claim.
(e) The student has access to but does not have to use all of the following equipment.
• Sphere Set 1: spheres of the same known mass with different known diameters
• Sphere Set 2: spheres of the same known diameter with different known masses
• A motion detector that can measure velocity as a function of time
i. Indicate two quantities that when graphed could be used to determine whether the diameter of the sphere affects the terminal speed.
Vertical axis: _______________ Horizontal axis: _______________
ii. Briefly describe how the quantities graphed could be used to determine the relationship between sphere diameter and terminal speed.
▶️Answer/Explanation
2(a)Example Solution
Σ F = ma
\(F_{g}\) – \(F_{drag}\) = ma
mg – bv = ma
\(m\frac{dv}{dt} = mg – bv\)
2(b)(i)Example Solution
2(b)(ii)Example Solution
For the times leading up to \(t_{T}\), the slope of the graph is positive which means that the magnitude of the drag force is still increasing. After \(t_{T}\), the slope of the graph is zero which means that the magnitude of the drag force is constant and equal to the downward gravitational force, which indicates that the net force is zero and that the sphere has reached a constant terminal velocity.
1(c)Example Response
The magnitude of the drag force at terminal speed does not change since the mass of the sphere is not changed and the drag force at terminal speed does not depend on the initial speed of the sphere.
2(d)(i)Example Solution
2(d)(ii)Example Solution
\(\left| F_{drag}\right| =bv\)
\(\frac{\left| F_{drag}\right|}{v} =b\)
slope = b
\(b =\frac{15N – 5N}{21m/s – 5m/s}\)
b = 0.625 kg s
2(e)(i)Example Response
Vertical axis: Terminal velocity
Horizontal axis: Diameter of the sphere
2(e)(ii)Example Response
The slope of the diameter vs terminal velocity graph can be used to determine if sphere diameter affects terminal velocity.
QUESTION
A student drops a cylinder of mass m from rest. The air exerts a drag force of magnitude $F_{drag}$ on the cylinder, as
drag shown in Figure 1. The student models the magnitude of the drag force as $F_{drag}=bv^{2}$ , where v is the speed of the
cylinder, and b is a positive constant with appropriate units.
(a) Derive, but do NOT solve, a differential equation that could be used to determine the speed v of the cylinder
as a function of time t. Express your answer in terms of given quantities and physical constants, as appropriate.
(b) The student correctly sketches the speed v of the cylinder as a function of time t, as shown in Figure 2.
i. Draw a vertical line on the sketch in Figure 2 to indicate the earliest time at which $F_{drag}$ on the cylinder
is equal to the magnitude of the weight of the cylinder. Label this time as$t_{1}$ on the time axis.
ii. Justify the location of $t_{1}$. Explicitly reference appropriate features of the sketch in Figure 2.
(c) Rather than dropping the cylinder from rest, the student throws the cylinder upward with a nonzero initial
speed. The cylinder is in the same orientation as when the cylinder was previously dropped. The student allows
the cylinder to fall toward the ground.
Indicate whether the magnitude of the cylinder’s maximum downward speed after being thrown upward would be
greater than, less than, or equal to the maximum speed $v_{max}$ in Figure 2.
_____ Greater than _____ Less than _____ Equal to
Briefly justify your answer.
(d) The student conducts an experiment to better understand the relationship between maximum speed $v_{max}$ and
mass. The student collects data to determine the maximum speed for cylinders dropped from rest, each with
the same physical size and shape but a different mass m. The student then graphs $v_{max}^{2}$ as a function of mass.
i. Draw the best-fit line for the data.
ii. Use the best-fit line to calculate an experimental value for B.
A student claims that the magnitude of the maximum speed of a cylinder dropped from rest depends on the length
of the cylinder. The student designs an experiment to collect data that can be used to provide evidence to support
the claim. The student drops cylinders with the orientation shown in Figure 3.
(e) The student has access to but does not have to use all of the following equipment.
• Cylinder Set 1: cylinders of the same known length with different known masses
• Cylinder Set 2: cylinders of the same known mass with different known lengths
• A motion detector that can measure velocity as a function of time
i. Indicate two quantities that when graphed could be used to determine whether the length of the cylinder affects the maximum speed.
Vertical axis: ______________ Horizontal axis: _______________
ii. Briefly describe how the quantities graphed could be used to determine the relationship between
cylinder length and maximum speed.
▶️Answer/Explanation
Ans:-
(a) For a multi-step derivation that includes Newton’s second law of motion
Example Solution
$\sum F=ma$
$F_{g} -F_{drag}=ma_{y}$
$mg-bv^{2}=ma_{y}$
$m\frac{dv}{dt} =mg-bv^{2}$
(b)(i) For a vertical line labeled t¹ at the approximate location at which the line becomes horizontal
(b)(ii) For relating $t_{1}$ to the time at which the velocity versus time graph is constant or the slope of the line is zero
Example Response
Because the sketched line is horizontal after t¹, the velocity is constant. If the velocity is
constant, then the acceleration is zero. Therefore, the net force is zero, which means that the
gravitational and drag forces are equal in magnitude.
(c) For selecting “Equal to” with an attempt at a relevant justification
Example Response
At its peak, the cylinder will have a speed of 0 m/s. Therefore, the cylinder would reach the
same vmax as if the student had dropped the cylinder from rest at that height. Because the
cylinder reached vmax from the initial drop, the two max speeds are equal.
(d)(i) For drawing an appropriate line of best fit that approximates the data
(d)(ii) For calculating a value for the slope of the line using two points on the best-fit line
Scoring Note: Using data points that fall on the best-fit line is acceptable.
Example Solution
(e)(i) For indicating that the length of the cylinder should be graphed
(e)(ii) For describing how the quantities graphed are related to the conclusions of the experiment
Example Response
The slope of the length vs. maximum velocity graph can be used to determine if length affects terminal velocity.
Question
A hollow glass sphere of radius 8.0 cm rotates about a vertical diameter with frequency 5 revolutions per second. A small wooden ball of mass 2.0 g rotates inside the sphere, as shown in the diagram above.
(a) Draw a free-body diagram indicating the forces acting on the wooden ball when it is at the position shown in the picture above.
(b) Calculate the angle \(\theta \), shown in the diagram above, to which the ball rises.
(c) Calculate the linear speed of the wooden ball as it rotates.
(d) The wooden ball is replaced with a steel ball of mass 20 g. Describe how the angle \(\theta \) to which the ball rises will be affected. Justify your answer.
Answer/Explanation
Ans:
(a) 1 pt: The weight of the ball acts down.
1 pt: The normal force acts up and left, perpendicular to the surface of the glass.
1 pt: No other forces act.
(b) 1 pt: The normal force can be broken into vertical and horizontal components, where the vertical is \(F_{N}\cos \theta\) and the horizontal is \(F_{N}\sin \theta \). (The vertical direction goes with cosine here because \(\theta\) is measured from the vertical.)
1 pt: The net vertical force is zero because the ball doesn’t rise or fall on the glass. Setting up forces equal to down, \(F_{N}\cos \theta =mg\).
1 pt: The horizontal force is a centripetal force, so \(F_{N}\sin \theta =mv^{2}/r\sin \theta \).
1 pt: For using \(r\sin \theta \) and not just r. (Why? Because you need to use the radius of the actual circular motion, which is not the same as the radius of the sphere.)
1 pt: The tangential speed “v” is the circumference of the circular motion divided by the period. Since period is 1/f, and because the radius of the circular motion is \(r\sin \theta \), this speed \(v=2\pi r\sin \theta f\).
1 pt: Now divide the vertical and horizontal force equations to get rid of the \(F_{N}\) term: \(\sin \theta /\cos \theta =v^{2}/r\sin \theta g\).
1 pt: Plug in the speed and the sin q terms cancel, leaving \(\cos \theta =g/4\pi ^{2}rf^{2}\).
1 pt: Plugging in the given values (including r = 0.08 m), q = 83°.
(c) 1 pt: From part (a), the linear speed is \(2\pi r\sin \theta f\).
1 pt: Plugging in values, the speed is 2.5 m/s. (If you didn’t get the point in part (a) for figuring out how to calculate linear speed, but you do it right here, then you can earn the point here.)
(d) 1 pt: The angle will not be affected.
1 pt: Since the mass of the ball does not appear in the equation to calculate the angle in part (b), the mass does not affect the angle.
Question
A block of mass m, which has an initial velocity vo at time t = 0, slides on a horizontal surface. If the sliding friction force f exerted on the block by the surface is directly proportional to its velocity (that is, f = -kv) determine the following:
a. The acceleration a of the block in terms of m, k, and v.
b. The speed v of the block as a function of time t.
c. The total distance the block slides.
Answer/Explanation
Ans:
a. F = ma; F = –kv = ma; a = –kv/m
b. \(\frac{dv}{dt}= \frac{kv}{m}\)
\(\frac{dv}{v}= – \frac{k}{m}dt\)
\(\int_{v_{0}}^{v}\frac{dv}{v}= \int_{0}^{t} – \frac{k}{m}dt\)
\(ln \left | \frac{v}{v_{0}} \right | = \frac{k}{m}t\)
\(\frac{v}{v_{0}} =e ^{-\frac{k}{m}t}\)
\(v = {v_{0}}^{e^{-\frac{k}{m}t}}\)
c. \(\frac{dx}{dt} = {v_{0}}^{e^{-\frac{k}{m}t}} \)
\(\int_{0}^{x}dx = \int_{0}^{\infty } {v_{0}}^{e^{-\frac{k}{m}t}dt}\)
\(x = -\frac{mv_{0}}{k}e^{-\frac{k}{m}t}\int_{0}^{\infty }\)
\(x = \frac{mv_{0}}{k}\)