Chapter 8 Rotational Dynamics and Static Equilibrium 8.1 Rotational motion 8.2 Torque 8.3 Equilibrium and Center of mass
Chapter 8.1 Rotational Motion The Physics of rotational motion is analogous to the physics of linear motion. For example, the new concept of Torque and angular acceleration are the rotational analogs of force and acceleration (F=ma), Newton’s II law.
Chapter 8.1 Rigid Body – an object whose size and shape do not change as it moves.
Chapter 8.1 3 basic types of motions of a rigid body: Translational motion – moving forward Rotational motion - rotate Combination motion – rotate and move forward
Draw trajectory of 3 motions.
Chapter 8.1 Translational (linear) Motion Rotational Motion Position – x [m] Displacement – Δx Angular position – θ [rad] Angular displacement - Δθ Velocity - 𝝊 [m/s] Angular velocity - 𝝎 [rad/s] Force – F [N] Torque - 𝞃 [Nm] Mass - m Moment of Inertia - I
Chapter 8.1 All points on a wheel rotate with the same angle θ. Because of that, every point on rotating rigid body has the same angular velocity 𝝎. arc or length A B
Practice problem The disk in a computer disk drive spin up to 5400 rpm in 2.00 s. What is the angular acceleration of the disk? At the end of 2.00 s, how many revolutions has the disk made?
11-9 The Vector Nature of Rotational Motion The direction of the angular velocity vector is along the axis of rotation. A right-hand rule gives the sign.
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Definition of Torque Torque is defined as the tendency to produce a change in rotational motion. Examples: Torque is a twist or turn that tends to produce rotation. Applications are found in many common tools around the home or industry where it is necessary to turn, tighten or loosen devices.
Torque is Determined by Three Factors: The magnitude of the applied force. The direction of the applied force. The location of the applied force. 20 N Location of force The forces nearer the end of the wrench have greater torques. 20 N Magnitude of force 40 N The 40-N force produces twice the torque as does the 20-N force. Each of the 20-N forces has a different torque due to the direction of force. 20 N Direction of Force q
Units for Torque Torque is proportional to the magnitude of F and to the distance r from the axis. t = Fr Units: Nm or lbft t = (40 N)(0.06 m) = 2.40 Nm, cw 6 cm 40 N t = 24.0 Nm, cw
Torque is a vector quantity that has direction as well as magnitude. Direction of Torque Torque is a vector quantity that has direction as well as magnitude. Turning the handle of a screwdriver clockwise and then counterclockwise will advance the screw first inward and then outward.
Direction of Torque Only 2 directions: counterclockwise torques are positive and clockwise torques are negative. ccw Positive torque: Counter-clockwise, out of page cw Negative torque: clockwise, into page
Line of Action of a Force The line of action of a force is an imaginary line of indefinite length drawn along the direction of the force. F2 F1 F3 Line of action
The Moment Arm The moment arm of a force is the perpendicular distance from the line of action of a force to the axis of rotation. F1 r F2 F3 r r
Only the tangential component of force causes a torque:
This leads to a more general definition of torque:
Torque = force x moment arm Calculating Torque Read problem and draw a rough figure. Extend line of action of the force. Draw and label moment arm. Calculate the moment arm if necessary. Apply definition of torque: Torque = force x moment arm
Extend line of action, draw, calculate r. Example 1: An 80-N force acts at the end of a 12-cm wrench as shown. Find the torque. Extend line of action, draw, calculate r. r = 12 cm sin 600 = 10.4 cm t = (80 N)(0.104 m) = 8.31 N m
Alternate: An 80-N force acts at the end of a 12-cm wrench as shown Alternate: An 80-N force acts at the end of a 12-cm wrench as shown. Find the torque. positive 12 cm Resolve 80-N force into components as shown. Note from figure: rx = 0 and ry = 12 cm t = (69.3 N)(0.12 m) t = 8.31 N m as before
11-9 The Vector Nature of Rotational Motion A similar right-hand rule gives the direction of the torque.
Conservation of angular momentum means that the total angular momentum around any axis must be constant. This is why gyroscopes are so stable.
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Calculating Resultant Torque Read, draw, and label a rough figure. Draw free-body diagram showing all forces, distances, and axis of rotation. Extend lines of action for each force. Calculate moment arms if necessary. Calculate torques due to EACH individual force affixing proper sign. CCW (+) and CW (-). Resultant torque is sum of individual torques.
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Chapter 8.3 Zero Torque and Static Equilibrium
Chapter 8.3 Static equilibrium occurs when an object is at rest – neither rotating nor translating.
Center of Mass The center of mass, also called the centroid or center of gravity, is the point of a body at which the force of gravity can be considered to act
The center of mass (CM) is the point where all of the mass of the object is concentrated. When an object is supported at its center of mass there is no net torque acting on the body and it will remain in static equilibrium
How to find the center of mass If the object is uniform, for example a meter stick, the center of mass will be at the exact geometric center.
If the object is irregular in shape, the center of mass is always located closer to the more massive end
This fact can be used to find the center of mass of an object – suspend it from different axes and trace a vertical line. The center of mass is where the lines meet.
The degree of stability in an object's position depends on how must its center of gravity will be changed if it is moved. Stable (It will be stable if the center of gravity lies below the pivot point. ) Unstable Neutral
remember the fun see-saw of our youth. See-Saws We all remember the fun see-saw of our youth. But what happens if . . .
Balancing Unequal Masses d1 d2 Need: M1 d1 = M2 d2
If an extended object is to be balanced, it must be supported through its center of mass.
The center of mass is closer to the more massive object. For two objects: The center of mass is closer to the more massive object.
Chapter 8.3: Equilibrium
Translational Equilibrium An object is said to be in Translational Equilibrium if and only if there is no resultant force. This means that the sum of all acting forces is zero. B A C In the example, the resultant of the three forces A, B, and C acting on the ring must be zero.
Conditions for Static Equilibrium
If the net torque is zero, it doesn’t matter which axis we consider rotation to be around; we are free to choose the one that makes our calculations easiest.
Solving static's problems Choose one object at a time and draw free- body diagram Choose a convenient coordinate system Write the equilibrium force equations and torque equation. Pay attention to determine the lever arm for each force correctly and a sign of torque. Solve these equations for unknown. Three equations are maximum of 3 unknowns to be solved for.
Practice problem Determine the mass of the monkey. In the process, also calculate T1, T2, T3, T4, T5, T6, and W2 if W1 equals 350 N.
CONCLUSION: Chapter 5A Torque