How is the understanding of the torques acting on a system used to predict changes in rotational motion?
How can the understanding of linear motion be applied to rotational motion?
How does the distribution of mass within a body affect its rotational motion?
Understandings (HL Only)
Understandings (HL Only)
Guidance:
How is the understanding of the torques acting on a system used to predict changes in rotational motion?
Task 1.7.1 Investigating Torque
Task | ||
Task 1. Balance Beam | Ash, Tawanda, Hema, Noah | Katherine, Milana, Adam, Nimrod |
Task 2. Door & Wrench | My-Lan, Izyan, Nimir, (Valentin) | (Audrey) Jeffery, Samarth, Oras, Maya |
Task 3. Center of Mass of Almost Any 2D Shape | Lily, Yaseen, Aryan | Oliver, Iyanuoluwa, Luka Muayed |
Force is what causes an object to accelerate in linear kinematics. Similarly, torque is what causes an angular acceleration. Torque is the measure of the force that can cause an object to rotate about an axis. Hence, torque can be defined as the rotational equivalent of linear force.
Rotation and Revolution
The term rotation usually refers to movement about an axis within the body, for example the rotation of the Earth every 24 hours. The rotation of a body does not affect its location.
Revolution usually concerns movement of a body around an exterior point, for example the Earth revolves around the Sun every year. (Some situations may be described as a rotation or a revolution.)
The axis of rotation is the line about which an object can rotate.
Extended Object and Rigid Bodies
An extended object has dimensions. Not a point. In A4, we look at extended objects which are able to rotate.
A rigid body is one whose shape does not deform significantly under the action of forces. The simplest everyday examples of rotation include a wheel and a door handle.
If an unbalanced force acts on the center of mass of a rigid body, then it will have linear acceleration but it will not rotate. All the bodies in (a) would have the same magnitude of acceleration. However, if the unbalanced force does not act on the center of mass, as in the examples in (b), the bodies will rotate as well as accelerate.
We can define the center of mass as the point on a body through which an unbalanced force can act without causing rotation.
Comparing Linear and Rotational Motion
Defining Torque
“Turning Effect”
F1 has no turning effect because its line of action is through the axis of rotation.
F3 has an effect and it depends on the perpendicular distance from the from the axis of rotation to the line of action of the force (r sin θ).
F2 has the biggest turning effect because its line of action is perpendicular to a line joining its point of application to the axis.
The ‘turning effect’ of a force, F, is known as its torque, τ , and it depends on the magnitude of the force and the perpendicular distance from the axis of rotation to the line of action of the force.
θ is the angle between the axis of rotation to the line of action of the force.
When there is no actual rotation, torque is sometimes called the moment of a force.
Torque (a vector quantity) has the SI unit Nm but note that it is not equivalent to the unit of energy (a scalar quantity), the joule, which is also Nm.
!
Couples
A couple produces no resultant force on an object, so there is no translational acceleration.
When applied to a couple, torque can be described as the sum of the moments produced by each of the forces in the couple
The magnitude of the torque provided by a couple is simply twice the magnitude of the torque provided by each of the two individual forces,
τ = 2Fr sin θ.
Interesting EE Idea: Bicycles, Forces, and Torque
θ is the angle between the axis of rotation to the line of action of the force.
Rotational Equilibrium
If an object remains at rest, or continues to move in exactly the same way, it is described as being in equilibrium. Translational equilibrium occurs when there is no resultant force acting on an object (Newton’s first law – Topic A.2), so that it remains stationary or continues to move with a constant velocity (that is, in a straight line at a constant speed). A similar definition applies to rotational motion:
Balance Beam
Resultant Torque
When more than one torque acts on a body the resultant (net) torque can be found by simple addition, but clockwise and anticlockwise torques will oppose each.
For example, when an object is acted upon by a 12 Nm clockwise torque and a 15 Nm anticlockwise torque, the resultant torque is (15 − 12) = 3 Nm anticlockwise.
5 N
3 m
4 N
3 m
2 m
Solving Problems with Multiple Moments
This example considers End B so the r’s are 0.8, 0.6, and 0.4 m from B.
Whether you consider End A or End B you should arrive at the same values.
Try it with End A so the r’s are 0.8, 0.2, and 0.4 m from A.
Measure r from the moment you are picking.
How can the understanding of linear motion be applied to rotational motion?
Angular Displacement
Equations of Motion for Rotational Motion
Graphs of Rotational Motion
240 rad
Relationship between Angular and Linear Acceleration
! NOT IN DATA BOOKLET
Relationship between Angular and Linear Acceleration
Time (s) | Angular velocity (rad/s) | Linear velocity (m/s) |
0.60 | 1.03 | 0.25 |
2.72 | 4.86 | 1.15 |
3.80 | 6.76 | 1.60 |
4.65 | 8.12 | 1.93 |
5.37 | 9.26 | 2.20 |
Prove that r = 23 ± 1 cm
Using a = ɑr
Difference between centripetal and tangential acceleration
Radius (cm) | Angular velocity (rad/s) | Acceleration (m/s2) | Theoretical Acceleration | % Difference |
12 | 15.6 | 31.6 | | |
6 | 15.3 | 14.5 | | |
3 | 15.1 | 6.9 | | |
What kind of acceleration did we calculate in this second demo?
Newton’s Second Law Applied to Angular Motion
Show that
Hint - Start with Newton’s Second Law of Motion
Moment of Inertia
How does the distribution of mass within a body affect its rotational motion?
Moment of Inertia for Different Shapes
Which of these will have the greatest moment of inertia for the same mass of the system?
If the mass of the object is concentrated near the axis of rotation, it’s relative rotational inertia will be smaller.
If the mass of the object is located further away from the system it’s relative rotational inertia will be larger.
The largest fraction will be the thin hoop because all of its mass is farthest from its axis of rotation.
Learn more about how it is done.
Defining Angular Momentum
Angular momentum, L, of a rotating object is the rotational equivalent of linear momentum (p = mv). It depends on the moment of inertia, I, of the object and its angular velocity (speed), ω.
Conservation of Angular Momentum
Similar to the law of conservation of linear momentum, the law of conservation of angular momentum (as follows) has no exceptions and can be used to predict changes to rotating systems.
Can you explain why the skater speeds up?
When a skater is spinning with arms outstretched, they have a relatively large moment of inertia because their mass is spread out from the center. According to the law of conservation of angular momentum, if no outside force acts to slow or speed up their spin, their total angular momentum stays the same.
When the skater pulls their arms in, they reduce their moment of inertia (their mass is now closer to the center). Since angular momentum must stay constant, and it equals moment of inertia multiplied by angular velocity, a decrease in the skater’s moment of inertia forces an increase in their spin speed (angular velocity). This is why pulling their arms in makes them spin faster.
Gyroscopic Precession
! Explanation out of syllabus
Angular Impulse
If the applied resultant torque changes, an average value should be used to determine an impulse.
Rotational Kinetic Energy
Rolling without Slipping
Rolling down a Slope
Rearrange for ⍵
Which will be first?
Understandings (HL Only)
Understandings (HL Only)