Two identical masses of magnitude $m$ are attached to the opposite ends of a solid beam of negligible mass and total length $l$. 

The beam is suspended from its centre and rotates about a vertical axis at constant angular speed $\omega$ as shown in the diagram below.

:::center
![PH1055xx](question:b5cad3c8-650b-42e2-8be4-d6b9db633e88)
:::

Which of the following correctly expresses the magnitude of the angular momentum of the system around its axis of rotation?

:br

::indent
- A. $\hspace{1em}  $ $\dfrac{1}{4}ml^2\omega$

- B. $\hspace{1em}  $ $\dfrac{1}{2}ml^2\omega$

- C. $\hspace{1em}  $ $ml^2\omega$

- D. $\hspace{1em}  $ $2ml^2\omega$


A point mass $m$ is attached to the end of a uniform beam with negligible mass and length $l$. The beam is placed horizontally with its other end hinged from a fixed point . 

:::center
![image](question:b897dd6b-cd2c-4cc3-9dde-dd1c4624353e)
:::

The system is released from rest and as it falls it rotates around point A.

Which of the following correctly describes the magnitude of the initial angular acceleration of the system?

:br

::indent
- A. $\hspace{1em}  $ $\dfrac{l}{g}$

- B. $\hspace{1em}  $ $\dfrac{2l}{g}$

- C. $\hspace{1em}  $ $\dfrac{g}{l}$

- D. $\hspace{1em}  $ $\dfrac{2g}{l}$

A solid sphere with radius $r$ and mass $m$ is rolling along a horizontal surface without slipping with an angular speed $\omega$ as shown in the diagram below.

:::center
![image](question:352a8eaa-f4f3-4eac-9cbf-21a008b42ebb)
:::

Which of the following correctly describes the magnitude of the linear momentum of the sphere?

:br

::indent
- A. $\hspace{1em}  $ $m\omega r$

- B. $\hspace{1em}  $ $m{\omega}^2r$

- C. $\hspace{1em}  $ $m\omega r^2$

- D. $\hspace{1em}  $ $\dfrac{m\omega}{r}$

A uniform solid beam with a length of $2.00\,m$ and a weight of $800\,N$ is placed horizontally and hinged at a wall at point O. It is supported by string AB which is attached to the beam at A and to the wall at B.

:::center
![image](question:c43913de-cae1-4a19-be33-e4cd74066025)
:::

Which of the following correctly describes the magnitude of the tension on the string?

:br

::indent
- A. $\hspace{1em}  $ $200\,N$

- B. $\hspace{1em}  $ $400\,N$

- C. $\hspace{1em}  $ $600\,N$

- D. $\hspace{1em}  $ $800\,N$

A wheel rotates from rest with uniform angular acceleration. At time $16\, s$, its angular velocity is 5 revolutions per second.

What is the angular displacement of the wheel during this time, in radians?

::indent
- A. $\hspace{1em}  20\pi\, $ 


- B. $\hspace{1em}  40\pi\,$ 


- C. $\hspace{1em}  80\pi\,$ 


- D. $\hspace{1em}  160\pi\,$ 


An object sits on a rotating horizontal disc of radius $r$. The disc rotates from rest with uniform angular acceleration $\alpha$. After time $t$ the object slides off of the disc.

What is the linear speed of the object at this time?

::indent
- A. $\hspace{1em}  \alpha t$ 


- B. $\hspace{1em}  r\alpha t$ 


- C. $\hspace{1em}  \dfrac{1}{2}\alpha t^2$ 


- D. $\hspace{1em}  \dfrac{1}{2}r\alpha t^2$ 


The graph below shows the angular momentum $L$ of a rotating solid object against time.

:::center
![image](question:16c015a8-c0a6-438b-a7c9-3dcadc4a2574)
:::

Which of the following correctly describes the gradient of the graph?

:br

::indent
- A. $\hspace{1em}  $ Angular acceleration

- B. $\hspace{1em}  $ Angular speed

- C. $\hspace{1em}  $ Moment of inertia

- D. $\hspace{1em}  $ Net torque

[Maximum mark: 9\]

In an amusement park, children enjoy a carousel ride that has a rotating platform and seats in the shape of horses on the platform. Some of the horses are closer to the centre of the platform. A top view of the ride is shown below.

:::center
![PH0615](question:a7fa9cb1-ea24-4da2-9b54-f5773b0d7a1f)
:::

:br

After accelerating to a certain angular velocity, the torque of the motor that rotates the carousel is equal to the torques of the opposing forces. 

1. The carousel makes 20 revolutions in $5.0 \, min$. Determine the angular velocity of the carousel. :marks[2]
    
2.  1. State the law of conservation of angular momentum. :marks[2]

    2. During the ride, some children on the outer horses move to horses that are closer to the centre. Explain the effect this could have on the angular velocity of the carousel. :marks[3]

3. Another carousel is moving with an angular velocity of $0.035 \, {rad\,s}^{-1}$. In an instant, the magnitude of the rate of change of moment of inertia due to a child moving towards the centre is $24 \, kg \, m^2 \, s^{-1}$. Determine the net torque acting on the carousel during the motion of the child. :marks[2]

A tightrope walker holds a long pole during his stunt.


:::center
![PH0705](question:871c6978-0c29-41b7-a982-8998112032d9)
$\footnotesize{\textrm{[Source: Image created using Images by  Mote Oo Education from Pixabay]}}$
:::


Three statements are given about the effect of the pole on the walker's ability to maintain their balance.

::indent
- I.	The pole increases the moment of inertia of the walker.

- II.	Use of the pole means that an unbalanced torque will produce a smaller angular acceleration to the walker and pole.

- III.	Holding the pole higher will make the walker more stable.



Which of these statements is/are correct?

::indent
- A. $\hspace{1em}  $ I and II only


- B. $\hspace{1em}  $ I and III only


- C. $\hspace{1em}  $ II and III only


- D. $\hspace{1em}  $ I, II and III

The graph shows the variation of torque on an object with time.

:::center
![PH0707](question:d9c4efbd-0b5e-45c1-a2b8-a6e45d0d2253)
$\footnotesize{\textrm{[Source: Image created using https://www.geogebra.org/calculator ]}}$


:::

The moment of inertia is $0.25\, kg\, m^2$.

Assuming the object starts from rest, what is the final angular speed?

::indent
- A. $\hspace{1em}  8\,rad\,s^{-1}$ 


- B. $\hspace{1em}  12\,rad\,s^{-1}$ 


- C. $\hspace{1em}  20\,rad\,s^{-1}$ 


- D. $\hspace{1em}  80\,rad\,s^{-1}$ 

A uniform beam of total length $L$ and mass $m$ is hinged on one end and acted upon by a horizontal force of magnitude $F$ on its other end.

The beam is held stationary at rest at an angle $\theta$ above the horizontal as shown in the diagram below.

:::center
![image](question:bee6f0aa-5f07-4794-b386-a4968a929255)
:::

Which of the following correctly expresses the value of $F$?

:br

::indent
- A. $\hspace{1em}  $ $2mg\tan{\theta}$

- B. $\hspace{1em}  $ $\dfrac{2mg}{\tan{\theta}}$

- C. $\hspace{1em}  $ $\dfrac{mg}{2\tan{\theta}}$

- D. $\hspace{1em}  $ $\dfrac{mg\tan{\theta}}{2}$

An object which is initially rotating with a frequency of $2\,s^{-1}$. It is acted upon a constant torque which produces an angular deceleration of magnitude $6\,\text{rad}\, s^{-2}$.

Which of the following best describes the number of complete rotations of the object until it comes to rest?

:br

::indent
- A. $\hspace{1em}  $ 2

- B. $\hspace{1em}  $ 3

- C. $\hspace{1em}  $ 4

- D. $\hspace{1em}  $ 6


\[Maximum mark: 10\]

Two solid cylinders with masses $3.5\, kg$ and $2.0\, kg$ are connected by a rigid rod of length $4.0\ m$ and negligible mass. The system rotates around a perpendicular axis $1.0\ m$ from the bigger mass as shown in the diagram.

:::center
![PH1066aaaa](question:0f18c542-3e7d-4745-9a19-87582de64a62)
:::

1. 1. Show that the moment of inertia of the system is approximately $22\, kg\, m^2$. :marks[1]  

    2. The system accelerates from rest with an angular acceleration of $4.0\, rad\, s^{-2}$.

        Calculate the kinetic energy of the system after $6.0\, s$. :marks[3]

2. 1. The $3.5\, kg$ cylinder is removed from the rod and released from rest to roll without slipping on a smooth inclined plane as shown in the diagram.

        :::center
        ![PH1066_Image2](question:96c0c04d-7f5d-4ada-970f-2c3a35ffac5c)
        :::

        The moment of inertia of the cylinder is $\dfrac{1}{2}\ mR^2$, where $R$ is the radius of the cylinder.

        Show that the angular velocity of the cylinder at the bottom of the incline is given by $\omega\ =\ \dfrac{2}{R}\ \sqrt{\dfrac{gh}{3}}$ :marks[3]

        2. The mass takes $4.2\, s$ to reach the bottom of the incline. If the radius of the cylinder is $22\, cm$, and the height of the incline is $1.5\ m$, calculate the average torque on the cylinder as it rolls down the ramp. :marks[3]

A uniform rod of mass $M$ and length $L$ rotates around a vertical axis through its center. Two equal forces act on the rod as shown in the diagram.

 
:::center
![Screenshot 2023-08-07 at 2.12.35 PM](question:643b69ac-8b0a-4838-b20d-66e33be9978b)
:::

The moment of inertia of the rod when rotating about its centre is  $\dfrac{1}{12} ML^2$.

Which of the following gives the angular acceleration of the rod?

::indent
- A. $\hspace{1em}  0$ 


- B. $\hspace{1em}  	\dfrac{6F}{ML}$ 


- C. $\hspace{1em}  	\dfrac{12F}{ML}$ 


- D. $\hspace{1em}  \dfrac{24F}{ML}$ 

\[Maximum mark: 20\]

1. In an Atwood machine, two blocks of masses $2.0\, kg$ and $4.0\, kg$ are connected with a massless string that passes over a pulley. The pulley has a radius $r$ of $0.25\, m$.
    :::center
    
    ![Screenshot 2023-07-29 at 10.17.33 AM](question:6aea9730-19d9-45e8-9f9a-463f72fd430f)
    :::
    The masses are released from rest and the tensions created on the strings attached to the masses of $2.0\, kg$ and $4.0\, kg$ are $T_1$ and  $T_2$ respectively. The acceleration of the mass of $2.0\, kg$ is $2.3\, m\,s^{-2}$. 
    1. Determine the values for $T_1$ and  $T_2$. :marks[3]
    
    2. Show that the angular acceleration of the pulley is approximately ${9\, rad\,s}^{-2}$. :marks[2]
    
    3. Calculate the angular speed of the pulley after $3.0\, s$.  :marks[2]
    
    4. By using the answer in (i) and the information in (ii), calculate the moment of inertia $I$ of the pulley. :marks[3]
    
    5. Discuss whether the pulley is in rotational equilibrium or translational equilibrium. :marks[3]

2. On another occasion, the same pulley\'s rotation is controlled by a motor. The variation of the pulley\'s angular velocity with time is shown on the graph below.
    :::center
    ![PH0613-B](question:d38a3e6e-3166-42a1-a6b4-7b1ea2523926)
    :::
    1. Identify the quantities that are represented by the area under the graph and the gradient of the graph. :marks[2]

    2. On the graph below, sketch the variation of torque $\Gamma$ on the pulley with time. (There is no need to add values to the axes.) :marks[2]

        :::center
        ![PH0613-B-ii](question:af2b3c8e-c50d-426d-ac9f-7691b10c23cb)
        :::
    
    3. The system starts from rest and the angular velocity of the pulley at $t=4.0\, s$ is $4.0\, rad{\, s}^{-1}$. Calculate the number of revolutions made by the pulley between $t=0\, s$ and $t=4.0\,s$. :marks[3]

The tip of the seconds hand of a clock moves from the $\text{\textquotedblleft}$12$\text{\textquotedblright}$ marking to the $\text{\textquotedblleft}$6$\text{\textquotedblright}$ marking with a tangential speed of $2 \,cm \, s^{-1}$.

Which of the following is correct about the angular speed and the centripetal acceleration?

:br

:::tableoptions{col1="hide"}
| |$\hspace{1em}$ Angular speed$\hspace{1em}$ | $\hspace{1em}$Centripetal acceleration $\hspace{1em}$|
| :-: | :-: | :-: |
| $\hspace{1em}$A.$\hspace{1em}$| $\dfrac{\pi}{60} \ rad\ s^{-1}$ | $\dfrac{\pi}{30} \ cm\ s^{-2}$ |
|$\hspace{1em}$B.$\hspace{1em}$| $\dfrac{\pi}{60} \ rad\ s^{-1}$ | $\dfrac{\pi}{15} \ cm\ s^{-2}$ |
|$\hspace{1em}$C.$\hspace{1em}$| $\dfrac{\pi}{30} \ rad{\ s}^{-1}$ | $\dfrac{\pi}{30} \ cm{\ s}^{-2}$ |
|$\hspace{1em}$D.$\hspace{1em}$| $\dfrac{\pi}{30} \ rad\ s^{-1}$ | $\dfrac{\pi}{15} \ cm{\ s}^{-2}$ |


[Maximum mark: 15\]

In a tricks competition, a snooker ball of radius 2.5 $cm$ is hit horizontally with a cue and rolls without slipping and then rolls up a slope. The mass of the ball is 0.16 $kg$ and the moment of inertia of the snooker ball is $\dfrac{2}{5} m r^2$. 

:::center
![PH0616](question:85dd9cc9-39c5-4691-8a51-51ad53324ee7)
:::

1. The average acceleration of the ball during the strike is 150$\,m\,s^{-2}$. Show that the average net torque applied on the ball during the strike is around 0.24$\,N\,m$. :marks[3]

2. The contact time of the cue to the ball is 2.5 $ms$. Show the total kinetic energy of the ball after 2.5$\,ms$ is approximately $1.6 \times{10}^{-2}\,J$. :marks[4]

3. 
    1. When the cue stops applying force on the ball, the ball starts to roll up an incline placed in front of the ball. Calculate the maximum vertical height reached by the ball. Ignore friction and air resistance. :marks[3]

    2. The angle of the incline is 25$\degree$ to the horizontal. What is the minimum coefficient of static friction between the surface and the ball for it to roll down the incline without slipping? :marks[5]

A disc of mass $m$ and radius $r$ rolls without slipping from rest down an inclined plane.

:::center
![PH0741](question:9dd73587-aca8-40a9-816d-afc4eaf1ea8a)
:::


 
The moment of inertia of the disc is $\dfrac{1}{2}mr^2$.

Which of the following is correct about the linear acceleration of the disc and the net torque?

:br


:::tableoptions{col1="hide"}
| | Linear Acceleration | Net Torque |
| :-: | :-: | :-: |
| $\hspace{1.5em}$A.$\hspace{1.5em}$| $\dfrac{2}{3}g\,sin \theta$  |$F_fr$ |
|$\hspace{1.5em}$B.$\hspace{1.5em}$ | $\dfrac{2}{3}g\,sin \theta$ | $mg\,sin \theta$ |
| $\hspace{1.5em}$C.$\hspace{1.5em}$ | $g\,sin \theta$|$F_fr$ |
| $\hspace{1.5em}$D.$\hspace{1.5em}$ | $g\,sin \theta$ | $mg\,sin \theta$ |


Two masses $M$ and $m$ are connected by a rod of negligible mass. They rotate around a vertical axis as shown in the diagram.

:::center
![PH0702](question:9ac1e239-268a-4d89-a71b-9bbd88185789)
:::

The ratio  

::indent
- $\dfrac{Distance\ between\ M\ and\ the\ axis}{Distance\ between\ m\ and\ the\ axis }=\dfrac{1}{2}$

What is the moment of inertia?

::indent
- A. $\hspace{1em}  	\dfrac{L^2 (M+m)}{4}$ 


- B. $\hspace{1em}  \dfrac{L^2 (M+4m)}{9}$ 


- C. $\hspace{1em}  L^2 (M+m)$ 


- D. $\hspace{1em}  L^2 (M+4m)$ 

An inextensible string with negligible mass is wound around a fixed pulley of moment of inertia $I$. The free end of the string is connected to a block of mass $M$ as shown in the figure. 

:::center
![image](question:d26cdfb3-c50f-4908-9465-e647f6f31586)
:::

The system is released from rest. Which of the following correctly describes the magnitude of the acceleration of the block?

:br

::indent
- A. $\hspace{1em}  $ $\dfrac{Mg}{M+\frac{I}{r^2}}$

- B. $\hspace{1em}  $ $\dfrac{Mg}{Mr^2+I}$

- C. $\hspace{1em}  $ $\dfrac{Mgr}{M+\frac{I}{r^2}}$

- D. $\hspace{1em}  $ $\dfrac{Mgr}{Mr^2+I}$


A student sits on a chair that is free to move. The student holds a rotating wheel as shown in the diagram.

:::center
![PH0740a](question:84d2afd6-4239-4c5a-a9ab-a34edb03b51b)
$\footnotesize{\textrm{[Source: Image created using Images by BedexpStock and OpenClipart-Vectors  from Pixabay]}}$

:::



From the student's perspective, the wheel rotates clockwise. Both the student and the chair are at rest.

The student inverted the direction of rotation of the wheel such that it is now turning anti-clockwise according to their perspective..

:::center
![PH0740b](question:cd5530d6-06c2-4269-bc06-31dc8b3bf218)
$\footnotesize{\textrm{[Source: Image created using Images by BedexpStock and OpenClipart-Vectors  from Pixabay]}}$
:::



Which of the following is correct about both the student and the chair system? 

::indent
- A. $\hspace{1em}  $It remains at rest to keep the total momentum zero. 


- B. $\hspace{1em}  $It rotates clockwise such that the total momentum is clockwise. 


- C. $\hspace{1em}  $It rotates counterclockwise such that the total momentum is clockwise. 


- D. $\hspace{1em}  $It rotates clockwise such that the total momentum is anticlockwise. 


A block with a mass $m_1$ is placed on a frictionless inclined plane. The plane is at an angle $\theta$ above the horizontal.

An inextensible string attached to his block is wound around a frictionless pulley with a radius $r$. 

A second pulley with a radius $2r$ rotates together with the first one. A block with a mass $m_2$ hangs from this pulley, attached to a different string.

:::center
![image](question:cdcd8cb9-f423-471b-9ef0-79bcfcb243a1)
:::

If the system is in equilibrium, which of the following correctly expresses ratio $\dfrac{m_1}{m_2}$?

:br

::indent
- A. $\hspace{1em}  $ $\dfrac{\sin{\theta}}{2}$

- B. $\hspace{1em}  $ $\dfrac{2}{\sin{\theta}}$

- C. $\hspace{1em}  $ $2\sin{\theta}$

- D. $\hspace{1em}  $ $\dfrac{1}{2\sin{\theta}}$

Disc $A$ rotates around a vertical frictionless axle with angular speed $\omega_A$ and moment of inertia $I_A$. Disc $B$ with moment of inertia $I_B$ is dropped from rest on disc $A$. The ratio  $\dfrac{I_B}{I_A} =\dfrac{1}{4}$

:::center
![PH0706](question:a028162c-8d96-41f0-81cc-323387729905)
:::

The two discs have the same final angular speed $\omega$.

What is the ratio  $\dfrac{(Final\ kinetic\ energy)}{(Initial\ kinetic\ energy)}$  ?

::indent
- A. $\hspace{1em}  \dfrac{1}{4}$ 


- B. $\hspace{1em}  \dfrac{1}{2}$ 


- C. $\hspace{1em}  \dfrac{4}{5}$ 


- D. $\hspace{1em}  \dfrac{5}{4}$ 

A uniform rod of mass $M$ and length $L$ is pivoted at one of its ends. The rod is released from rest from the horizontal position.

:::center
![PH0704](question:6324fe32-7fdd-4434-95f2-43c204f80c5a)
:::

The moment of inertia of the rod when rotating about its end is  $\dfrac{1}{3}ML^2$.

What is the linear speed of the bottom end of the rod as it swings past the vertical position?

::indent
- A. $\hspace{1em}  	\sqrt{\dfrac{3g}{L}}$ 


- B. $\hspace{1em}  	\sqrt{3gL}$ 


- C. $\hspace{1em}  	\sqrt{6gL}$ 


- D. $\hspace{1em}  \dfrac{3g}{2L}$ 


Torque, Angular Speed, Angular Acceleration, Moment of Inertia, Angular Momentum, Rotational Kinetic Energy

HL - 2025

Space, Time & Motion

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Displacement, Velocity, Acceleration, Uniform Acceleration Equations, Projectile Motion

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