Below is our AP Physics C unit 2 practice test. It covers Newton’s laws of motion and their application to solving problems with the help of free-body diagrams. Mathematical difficulty needed to solve the equations comprise of basic algebra, trigonometry, and a bit of calculus (Integration, differentiation and solving simple differential equations).
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Question 1 |
Two independent sources of force act on an object. The vector form of these forces is given below:
$\overrightarrow{F}_I = 2 \skew{2.5}\hat{i} + 7 \skew{4}\hat{j} \ N$ and $\overrightarrow{F}_{II} = 5 \skew{2.5}\hat{i} + 4 \skew{4}\hat{j} \ N$
What is the acceleration of the object, in $m/s^2$, if its mass is $4 \ kg$?
$ \dfrac{3}{4} (\skew{2.3}\hat{i} + \skew{4.5}\hat{j} ) $ | |
$ \dfrac{1}{6} (12 \skew{2.3}\hat{i} +10 \skew{4.5}\hat{j} )$ | |
$ \dfrac{1}{4} (11 \skew{2.3}\hat{i} +7 \skew{4.5}\hat{j} )$ | |
$ \dfrac{1}{4} (6 \skew{2.3}\hat{i} +9 \skew{4.5}\hat{j} ) $ | |
$ \dfrac{1}{4} (7 \skew{2.3}\hat{i} +11 \skew{4.5}\hat{j} )$ |
Question 1 Explanation:
Net force acting on the object
$=(2 \skew{2}\hat{i} +7 \skew{5}\hat{j} )+(5\skew{2}\hat{i}+4 \skew{5}\hat{j} )$
$=(7 \skew{2}\hat{i} +11 \skew{5}\hat{j} ) $
Acceleration of the object
$=\dfrac{1}{mass} × Force$
$=\dfrac{1}{4} (7 \skew{2}\hat{i} +11 \skew{5}\hat{j} ) $
$=(2 \skew{2}\hat{i} +7 \skew{5}\hat{j} )+(5\skew{2}\hat{i}+4 \skew{5}\hat{j} )$
$=(7 \skew{2}\hat{i} +11 \skew{5}\hat{j} ) $
Acceleration of the object
$=\dfrac{1}{mass} × Force$
$=\dfrac{1}{4} (7 \skew{2}\hat{i} +11 \skew{5}\hat{j} ) $
Question 2 |
Look at the below Atwood Machine with a frictionless pulley. Assume the string and the pulley are of negligible mass.
![](https://highschooltestprep.com/wp-content/uploads/2024/03/Quiz219-Question2.jpg)
The system is released from rest. The acceleration of the $2 \ kg$ block is,
$2.5 \ m/s^2$ | |
$2.0 \ m/s^2$ | |
$3.6 \ m/s^2$ | |
$4.3 \ m/s^2$ | |
$5.4 \ m/s^2$ |
Question 2 Explanation:
The free body diagram of the two blocks is,
![](https://highschooltestprep.com/wp-content/uploads/2024/03/Quiz219-Explanation2.jpg)
$T-2g=2a$ … (1)
$3g-T=3a$ … (2)
Adding (1) and (2),
$g = 5a → a = \dfrac{g}{5} =2 \ m/s^2 $
![](https://highschooltestprep.com/wp-content/uploads/2024/03/Quiz219-Explanation2.jpg)
$T-2g=2a$ … (1)
$3g-T=3a$ … (2)
Adding (1) and (2),
$g = 5a → a = \dfrac{g}{5} =2 \ m/s^2 $
Question 3 |
A frictionless circular shaped ring has a small marble of mass $m$ fixed on it, as shown below.
![](https://highschooltestprep.com/wp-content/uploads/2024/03/Quiz219-Question3.jpg)
The marble is set into motion at a uniform speed $u$. Which of the below forces keeps the marble moving at a uniform speed?
Friction between marble and the ring | |
Weight of the marble | |
Weight of the ring | |
Normal force due to the ring | |
None of the above |
Question 3 Explanation:
Check each option,
Option A → Incorrect, as friction would slow down the marble in some time
Option B → Incorrect, as the weight of the marble is perpendicular to the plane of the velocity
Option C → Incorrect, as the weight of the ring cannot affect the motion of the marble directly
Option D → Correct, as the normal force is always perpendicular to the velocity of the marble and is in the same plane
Option E → Incorrect, as D is correct
Option A → Incorrect, as friction would slow down the marble in some time
Option B → Incorrect, as the weight of the marble is perpendicular to the plane of the velocity
Option C → Incorrect, as the weight of the ring cannot affect the motion of the marble directly
Option D → Correct, as the normal force is always perpendicular to the velocity of the marble and is in the same plane
Option E → Incorrect, as D is correct
Question 4 |
A block of mass $2 \ kg$ is on an inclined plane. The coefficient of static friction between the block and the plane is $0.4$. How much force would be needed to set the block in motion (down the incline) if the angle of the inclined plane is $12°$?
$3.67 \ N$ | |
$7.82 \ N$ | |
$5.22 \ N$ | |
$4.15 \ N$ | |
$2.33 \ N$ |
Question 4 Explanation:
In equilibrium, the free body diagram of the block is as follows:
![](https://highschooltestprep.com/wp-content/uploads/2024/03/Quiz219-Explanation4.jpg)
For equilibrium,
$N = 2g \cos 12°$ and $f_r = 2g \ \sin 12°$
Using $g = 10 \ m/s^2,$
$N = 19.56 \ N$ and $f_r = 4.15 \ N$
In static condition, $f_r ≤ μN$ (which is true for the system)
Extra force needed to set the block in motion (down the incline)
$=μN-f_r$
$=0.4 × 19.56 - 4.15$
$=7.82 - 4.15$
$=3.67 \ N$
![](https://highschooltestprep.com/wp-content/uploads/2024/03/Quiz219-Explanation4.jpg)
For equilibrium,
$N = 2g \cos 12°$ and $f_r = 2g \ \sin 12°$
Using $g = 10 \ m/s^2,$
$N = 19.56 \ N$ and $f_r = 4.15 \ N$
In static condition, $f_r ≤ μN$ (which is true for the system)
Extra force needed to set the block in motion (down the incline)
$=μN-f_r$
$=0.4 × 19.56 - 4.15$
$=7.82 - 4.15$
$=3.67 \ N$
Question 5 |
Two blocks are connected by a massless string of length $l$. A force of $25 \ N$ acts on the right block, as shown below.
![](https://highschooltestprep.com/wp-content/uploads/2024/03/Quiz219-Question5.jpg)
If the coefficient of friction for the right block is $0.33$, then what is the value of the coefficient of friction $μ$ for the left block? (Assume the system remains just in equilibrium and any small force can set the system in motion.)
$0.250$ | |
$0.333$ | |
$0.223$ | |
$0.252$ | |
$0.283$ |
Question 5 Explanation:
The free body diagram for the two-block system is given below:
![](https://highschooltestprep.com/wp-content/uploads/2024/03/Quiz219-Explanation5.jpg)
For the right block: $25 = T + f_1$ and $N = 5g =50$
Combining the two equations:
$25 = T +0.33 × 50$
$T = 8.5 \ N$
For the left block: $T = f_2$ and $N = 3g = 30$
Combining the two equations:
$8.5 = 30μ$
$μ = 0.283$
![](https://highschooltestprep.com/wp-content/uploads/2024/03/Quiz219-Explanation5.jpg)
For the right block: $25 = T + f_1$ and $N = 5g =50$
Combining the two equations:
$25 = T +0.33 × 50$
$T = 8.5 \ N$
For the left block: $T = f_2$ and $N = 3g = 30$
Combining the two equations:
$8.5 = 30μ$
$μ = 0.283$
Question 6 |
Two blocks are kept on each other, as shown below.
![](https://highschooltestprep.com/wp-content/uploads/2024/03/Quiz219-Question6.jpg)
The coefficient of static friction between the two blocks is $0.25$, and the coefficient of kinetic friction is $0.18$. If a force of $40 \ N$ acts on the lower block, what will be the acceleration of the upper block? (Assume there is no friction between the lower block and the floor)
$2.25 \ m/s^2$ | |
$2.85 \ m/s^2$ | |
$1.80 \ m/s^2$ | |
$1.55 \ m/s^2$ | |
$1.20 \ m/s^2$ |
Question 6 Explanation:
Assuming the two blocks move together with the same acceleration a,
$a = \dfrac{F}{m_{down} + m_{up}}$
$=\dfrac{40}{14}$
$=2.85 \ m/s^2$
The free body diagram for the upper block is:
Thus, for the block to move with an acceleration of $2.85 \ m/s^2$ the frictional force must be at least $4 × 2.85 = 11.4 \ N$
Maximum value of static friction $= 0.25 × 40 = 10 N < 11.4 \ N$
Thus, the upper block slides with acceleration lesser than the lower block
$a = \dfrac{μ_k \ N}{m}$
$= \dfrac{0.18×40}{4} $
$= 1.8 \ m/s^2 $
$a = \dfrac{F}{m_{down} + m_{up}}$
$=\dfrac{40}{14}$
$=2.85 \ m/s^2$
The free body diagram for the upper block is:
![](https://highschooltestprep.com/wp-content/uploads/2024/03/Quiz219-Explanation6.jpg)
Thus, for the block to move with an acceleration of $2.85 \ m/s^2$ the frictional force must be at least $4 × 2.85 = 11.4 \ N$
Maximum value of static friction $= 0.25 × 40 = 10 N < 11.4 \ N$
Thus, the upper block slides with acceleration lesser than the lower block
$a = \dfrac{μ_k \ N}{m}$
$= \dfrac{0.18×40}{4} $
$= 1.8 \ m/s^2 $
Question 7 |
A particle of mass $m$ is moving through a horizontally oriented medium. The particle experiences a drag force of $γv$ in a direction opposite to the velocity. If there is no other force acting on the particle, then which of the below equations represents the motion of the particle?
$m \dfrac{dv}{dt} - γv = 0$ | |
$m \dfrac{dv}{dt} + γv = 0$ | |
$m \dfrac{dv}{dt} - γv = ma$ | |
$m \dfrac{dv}{dt} + 2γv = 0$ | |
None of the above |
Question 7 Explanation:
$F = m \dfrac{dv}{dt}$
The only force acting on the particle is the drag force of the form $-γv$
Thus, $m \dfrac{dv}{dt} = -γv$
Or $m \dfrac{dv}{dt} + γv = 0$
The only force acting on the particle is the drag force of the form $-γv$
Thus, $m \dfrac{dv}{dt} = -γv$
Or $m \dfrac{dv}{dt} + γv = 0$
Question 8 |
A stone of mass $m$ tied to a massless string of length $r$ is set into motion. If the stone moves in a vertical circle, what is the minimum velocity that the stone must have when it is at the topmost point of the circle?
$\sqrt{5gr}$ | |
$\sqrt{4gr}$ | |
$\sqrt{3gr}$ | |
$\sqrt{2gr}$ | |
$\sqrt{gr}$ |
Question 8 Explanation:
For minimum velocity, Tension force $= 0 \ N$
At the topmost point, the centripetal force is provided by the gravitational force of the stone
$mg = \dfrac{mv^2}{r} → v = \sqrt{rg} $
At the topmost point, the centripetal force is provided by the gravitational force of the stone
$mg = \dfrac{mv^2}{r} → v = \sqrt{rg} $
Question 9 |
The velocity vector of a car of mass m is given by the equation:
$\overrightarrow{v}(t) = 5t \skew{2.5}\hat{i} + (10 -t^2) \skew{4}\hat{j}$
What is the average force acting on the car from $t = 1$ to $t = 4$?
$2m(\skew{2.3}\hat{i} + \skew{4.5}\hat{j} )$ | |
$5m(\skew{2.3}\hat{i} - \skew{4.5}\hat{j} )$ | |
$4m(\skew{2.3}\hat{i} - \skew{4.5}\hat{j})$ | |
$m(2\skew{2.3}\hat{i} - 3 \skew{4.5}\hat{j} )$ | |
$3m(\skew{2.3}\hat{i} - \skew{4.5}\hat{j})$ |
Question 9 Explanation:
$\overrightarrow{v}(t = 1)=5 \skew{2}\hat{i} + 9 \skew{5}\hat{j}$ and $\overrightarrow{v}(t=4)=20 \skew{2}\hat{i} -6 \skew{5}\hat{j}$
Average acceleration
$=\dfrac{\overrightarrow{v}(t=4) - \overrightarrow{v}(t=1)}{4 - 1}$
$=\dfrac{15 \skew{2}\hat{i} - 15 \skew{5}\hat{j} }{3}$
$=5 \skew{2}\hat{i}-5 \skew{5}\hat{j}$
Average force $= m × \overrightarrow{a}=5m( \skew{2}\hat{i} - \skew{5}\hat{j} )$
Average acceleration
$=\dfrac{\overrightarrow{v}(t=4) - \overrightarrow{v}(t=1)}{4 - 1}$
$=\dfrac{15 \skew{2}\hat{i} - 15 \skew{5}\hat{j} }{3}$
$=5 \skew{2}\hat{i}-5 \skew{5}\hat{j}$
Average force $= m × \overrightarrow{a}=5m( \skew{2}\hat{i} - \skew{5}\hat{j} )$
Question 10 |
Questions 10, 11 and 12 are based on the below information:
Look at the below system of three blocks.
![](https://highschooltestprep.com/wp-content/uploads/2024/03/Quiz219-Question10-11-12.jpg)
The strings and the pulley are massless and the frictional force between the blocks and the surface is zero. When the system starts from rest, the tension force between the $4 \ kg$ block and the $2 \ kg$ block is,
$20 \ N$ | |
$12 \ N$ | |
$15 \ N$ | |
$10 \ N$ | |
$8 \ N$ |
Question 10 Explanation:
The entire system moves with the same magnitude of acceleration
For the block moving vertically: $2g - T = 2a$
For the two-block system moving horizontally: $T = 6a$
Using the two equations, $2g - 6a = 2a → a = \dfrac{g}{4}= 2.5 \ m/s^2$
For the 4 kg block, the equation of motion is:
$T_1 = 4a → T_1 = 10 \ N $
For the block moving vertically: $2g - T = 2a$
For the two-block system moving horizontally: $T = 6a$
Using the two equations, $2g - 6a = 2a → a = \dfrac{g}{4}= 2.5 \ m/s^2$
For the 4 kg block, the equation of motion is:
$T_1 = 4a → T_1 = 10 \ N $
Question 11 |
Look at the below system of three blocks.
![](https://highschooltestprep.com/wp-content/uploads/2024/03/Quiz219-Question10-11-12.jpg)
What is the acceleration of the $2 \ kg$ block kept on the table?
$2.5 \ m/s^2$ | |
$2.0 \ m/s^2$ | |
$3.3 \ m/s^2$ | |
$4.2 \ m/s^2$ | |
$1.8 \ m/s^2$ |
Question 11 Explanation:
From question 10,
$a = \dfrac{g}{4}$
Thus, the acceleration of the system of blocks is equal to
$2.5 \ m/s^2 $
$a = \dfrac{g}{4}$
Thus, the acceleration of the system of blocks is equal to
$2.5 \ m/s^2 $
Question 12 |
Look at the below system of three blocks.
![](https://highschooltestprep.com/wp-content/uploads/2024/03/Quiz219-Question10-11-12.jpg)
At $t=t_{break}$, the string between the $4 \ kg$ block and the $2 \ kg$ block breaks. What will be the acceleration of the $2 \ kg$ block now?
$3 \ m/s^2$ | |
$4 \ m/s^2$ | |
$5 \ m/s^2$ | |
$6 \ m/s^2$ | |
None of the above |
Question 12 Explanation:
After $t=t_{break}$,
For the block moving vertically: $2g-T=2a$
For the one-block system moving horizontally: $T=2a$
Using the two equations, $2g-2a=2a→a=\dfrac{g}{2}=5 \ m/s^2$
For the block moving vertically: $2g-T=2a$
For the one-block system moving horizontally: $T=2a$
Using the two equations, $2g-2a=2a→a=\dfrac{g}{2}=5 \ m/s^2$
Question 13 |
A block at the bottom of an inclined plane is given a sharp push at $t = 0 \ s$.
![](https://highschooltestprep.com/wp-content/uploads/2024/03/Quiz219-Question13.jpg)
As a result, the block moves upwards and travels a distance $h$ before it starts sliding down. The coefficient of friction between the block and the inclined plane is $0.15$, and the angle of inclination is $25°$. What is the value of $h$?
$34.56 \ m$ | |
$30.60 \ m$ | |
$32.04 \ m$ | |
$29.03 \ m$ | |
$22.16 \ m$ |
Question 13 Explanation:
The free body diagram of the block is:
![](https://highschooltestprep.com/wp-content/uploads/2024/03/Quiz219-Explanation13.jpg)
The equations of motion are:
$N=g \ \cos 25°$ and $g \ \sin 25° +f_r=a$
Using $f_r=0.15×N$ in the above equations gives,
$a=g \sin 25° +0.15g \ \cos 25° $
$a=g(\sin 25° +0.15 \ \cos 25° )$
$a=5.58 \ m/s^2$
Using $ a=5.58 \ m/s^2, u=18 \ m/s$ and $v=0 \ m/s$ in $v^2=u^2-2ah$ gives,
$h=\dfrac{u^2}{2a} = \dfrac{18^2}{2×5.58}$
$h=29.03 \ m$
![](https://highschooltestprep.com/wp-content/uploads/2024/03/Quiz219-Explanation13.jpg)
The equations of motion are:
$N=g \ \cos 25°$ and $g \ \sin 25° +f_r=a$
Using $f_r=0.15×N$ in the above equations gives,
$a=g \sin 25° +0.15g \ \cos 25° $
$a=g(\sin 25° +0.15 \ \cos 25° )$
$a=5.58 \ m/s^2$
Using $ a=5.58 \ m/s^2, u=18 \ m/s$ and $v=0 \ m/s$ in $v^2=u^2-2ah$ gives,
$h=\dfrac{u^2}{2a} = \dfrac{18^2}{2×5.58}$
$h=29.03 \ m$
Question 14 |
A time-dependent centripetal force acts on an object of mass $2.4 \ kg$. The expression for the centripetal force is given as,
$\overrightarrow{f}_{cent} (t)=10[\cos 8t \skew{2.5}\hat{i} + \sin 8t \skew{4}\hat{j} ] N$
The object moves in a circle of radius 3 cm. Find the speed of the object when $t = 6 \ s$. (Note: The angles are in radians)
$0.980 \ m/s$ | |
$0.060 \ m/s$ | |
$0.802 \ m/s$ | |
$2.451 \ m/s$ | |
$0.353 \ m/s$ |
Question 14 Explanation:
$\overrightarrow{f}_{cent} (t=6)=10[\cos \ 48 \skew{2}\hat{i} + \sin \ 48 \skew{5}\hat{j} ]$
Thus, the magnitude of the centripetal force is
$\sqrt{10^2 (\sin^2 \ 48 + \cos^2 \ 48)} =10 \ N$
Since, $F = \dfrac{mv^2}{r} → v = \sqrt{ \dfrac{Fr}{m}}$
$v = \sqrt{10 × \dfrac{0.03}{2.4}}=0.353 \ m/s $
Thus, the magnitude of the centripetal force is
$\sqrt{10^2 (\sin^2 \ 48 + \cos^2 \ 48)} =10 \ N$
Since, $F = \dfrac{mv^2}{r} → v = \sqrt{ \dfrac{Fr}{m}}$
$v = \sqrt{10 × \dfrac{0.03}{2.4}}=0.353 \ m/s $
Question 15 |
The differential equation of an object moving through a viscous medium is given as:
$m \dfrac{dv}{dt} = -Av^3$
Where $A$ is a positive constant and $v(t)$ is the time-dependent velocity of the object. Find an expression for $v(t$) subject to the condition that $v(0)=1 \ m/s$.
$\left(1+\dfrac{2At}{m}\right)^{-1/2}$ | |
$\left(1+\dfrac{2At}{m}\right)^{-2}$ | |
$\left(1+\dfrac{2At}{m}\right)^{-1}$ | |
$\left(1+\dfrac{4At}{m}\right)^{1/2}$ | |
$\left(1+\dfrac{4At}{m}\right)^{2}$ |
Question 15 Explanation:
$m \dfrac{dv}{dt} = -Av^3$
Using separation of variables,
$ \dfrac{dv}{v^3} = -\dfrac{A}{m} \ dt$
Integrating both sides gives,
$\int \dfrac{dv}{v^3} =-\int \dfrac{A}{m} \ dt$
$\dfrac{v^{-2}}{-2} = - \dfrac{A}{m} t-c$ (Where $c$ is some constant)
$v^{-2} = \dfrac{2A}{m} t+c'$ (Where $c'$ is some other constant)
Using $v(t=0)=1 \ m/s$ gives,
$c'=1$
Thus, $v(t) = \left(1 + \dfrac{2At}{m}\right)^{-1/2}$
Using separation of variables,
$ \dfrac{dv}{v^3} = -\dfrac{A}{m} \ dt$
Integrating both sides gives,
$\int \dfrac{dv}{v^3} =-\int \dfrac{A}{m} \ dt$
$\dfrac{v^{-2}}{-2} = - \dfrac{A}{m} t-c$ (Where $c$ is some constant)
$v^{-2} = \dfrac{2A}{m} t+c'$ (Where $c'$ is some other constant)
Using $v(t=0)=1 \ m/s$ gives,
$c'=1$
Thus, $v(t) = \left(1 + \dfrac{2At}{m}\right)^{-1/2}$
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