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Torque and Rotational Motion AP Physics 1 Angular Kinematics Discussion Questions • How do we get around? • How do we move things? Angular Displacement • We’ve discussed linear motion, but what happens when things roll or rotate? • If we look at a wheel, how far will it travel linearly in one rotation? • Imagine we took the tire off the wheel and stretched it out long! Angular Displacement • When referring to how many times something rotates, we call it angular displacement or the change in the angle theta • Angular displacement, θ • Measured in radians • How do we convert from radians to meters? • 𝑥 = 𝑟∆𝜃 Δθ r Angular Displacement • A disk rotates 5 times. How many radians is this? 2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 5 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑠 × = 10 π 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 ≈ 30 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 1 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 • If the disk has a radius of .3 m, how many meters has it rotated through? 𝑥 = 𝑟∆𝜃 𝑥 = 0.3𝑚 ∗ 10𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 = 3𝜋 𝑚 ≈ 9 𝑚 OR 2𝜋 (.3 𝑚) 5 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑠 × = 3π 𝑚 ≈ 9 𝑚 1 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 Angular Velocity • Just like linear velocity is the rate of change of displacement, angular velocity it the rate of change of angular displacement. ∆𝜃 𝜔= ∆𝑡 • Angular velocity is represented by the lower case Greek letter omega and is measured in radians per second. Angular Velocity • We know a relationship exists between linear displacement and angular displacement, so it there a relationship between linear velocity and angular velocity? YES! • If a wheel with a radius of 0.5 m travels 3 meters in 1 second we know it’s linear velocity is 3 m/s. We also know that the linear distance traveled is directly related to the circumference of the wheel. 3𝑚 1 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 1 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 ∗ ∗ ∗ = 6 𝑟𝑎𝑑/𝑠𝑒𝑐 𝑠 1 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 2𝜋 .5 𝑚 1 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 𝑣 = 𝑟𝜔 Angular Velocity • A disk rotates 5 times in 2 seconds. What is it’s angular velocity? 2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 5 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑠 × = 10 π 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 ≈ 30 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 1 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 ∆𝜃 10 π 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 𝑟𝑎𝑑 𝜔= = =5𝜋 ∆𝑡 2 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 𝑠𝑒𝑐 • What is this in meters per second if the disk has a radius of 0.3 m? 𝑟𝑎𝑑 𝑚 𝑣 = 𝑟𝜔 = 0.3𝑚 ∗ 5 𝜋 = 1.5𝜋 𝑠𝑒𝑐 𝑠 • This assumes that the disk is rotating at a constant rate, but what happens if it is not rotating at a constant angular velocity? Angular Acceleration • Angular acceleration is the rate of change in angular velocity with respect to time just like linear acceleration is the rate of change in linear velocity with respect to time. ∆𝜔 𝛼= ∆𝑡 • It is represented by the lower case Greek letter alpha and is measured in radians per second squared. Angular Acceleration • Just as with the previous two quantities, we can easily convert between angular acceleration and linear acceleration. It even follows the same relationship! 𝑎 = 𝑟𝛼 Angular Acceleration • When there is angular acceleration, the angular displacement and angular velocity is effected in the same way that linear acceleration effects linear displacement and linear velocity. 1 2 𝜃 = 𝜃0 + 𝜔0 𝑡 + 𝛼𝑡 2 𝜔 = 𝜔0 + 𝛼𝑡 𝜔2 = 𝜔02 + 𝛼𝑡 Angular Acceleration A disk with a radius of 0.2 m starts at rests and uniformly accelerates to 10 rad/sec in 4 seconds. • What is the disk’s angular acceleration? 𝛼 = ∆𝜔 ∆𝑡 = 10−0 4 𝑟𝑎𝑑 𝑠𝑒𝑐 2 𝑚 0.5 2 𝑠 = 2.5 • What is the disk’s linear acceleration? 𝑎 = 𝑟𝛼 = .2 ∗ 2.5 = • How many times has the disk rotated in 4 seconds? 1 2 1 𝜃 = 𝜃0 + 𝜔0 𝑡 + 𝛼𝑡 = 0 + 0 + 2.5 42 = 20 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 2 2 1 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 10 20 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 ∗ = 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑠 2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 𝜋 Comparison of linear and angular quantities Linear Quantity Angular Quantity Displacement, x Angular Displacement, θ ∆𝑥 Velocity, 𝑣𝑥 = ∆𝑡 ∆𝑣𝑥 Acceleration, 𝑎𝑥 = ∆𝑡 ∆𝜃 Angular Velocity, 𝜔 = ∆𝑡 ∆𝜔 Angular Acceleration, 𝛼 = ∆𝑡 Converting between linear and angular quantities 𝑥 = 𝑟∆𝜃 𝑣 = 𝑟𝜔 𝑎 = 𝑟𝛼 Comparison of linear and angular kinematic formulas Linear Quantity Angular Quantity 1 𝑥 = 𝑥0 + 𝑣𝑥0 𝑡 + 𝑎𝑥 𝑡 2 2 1 2 𝜃 = 𝜃0 + 𝜔0 𝑡 + 𝛼𝑡 2 𝑣𝑥 = 𝑣𝑥0 + 𝑎𝑥 t 𝜔 = 𝜔0 + 𝛼𝑡 2 𝑣𝑥 = 2 𝑣𝑥0 +2𝑎𝑥 (𝑥 − 𝑥0 ) 2 𝜔 = 2 𝜔0 + 𝛼𝑡 Torque Discussion Questions • What causes things to rotate? • How do you get things to rotate faster or slower? Torque • • • • • • • • • What is a force? A torque is a twist! 𝜏 = 𝑟 × 𝐹 r is the distance between the axis of rotation and the location of the force. x is a cross product. A cross product is similar to the dot product that we saw in work, except that now you take the sine instead of the cosine. 𝜏 = 𝑟⊥ 𝐹 = 𝑟𝐹 sin 𝜃 Torque is represented by the lower case Greek letter Tau and is measured in NewtonMeters Torque is a vector quantity however the direction is not up, down, left, or right! It is clockwise or counterclockwise since torques cause rotations. (Counterclockwise is conventionally positive) It is the rotational analogue of force. Torque • If the board to the right is 3 meters long and a force of F=5N is applied at r=2.8 m. What is the torque on the board? • 𝜏 = 2.8 × 5 = 14 𝑁𝑚 𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒 r F Torque F • If the board to the right is 3 meters long and a force of F=5N is applied at r=2.8 m. What is the torque on the board? • 𝜏 = 2.8 ∗ 5 sin 45° = 9.9 𝑁𝑚 𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒 r 45° Torque • What happens when multiple forces act on an object? FN Fa • What happens when multiple torques act on an object? • When dealing with multiple torques, make sure it is clear where your axis of rotation is since this is how r is measured. r Ff Fg Equilibrium • Is the box below in translational equilibrium? • YES! 5N • Is it in rotational equilibrium? • NO! It will rotate clockwise. 5N Torque and Static Equilibrium • We know a system is in translational equilibrium when the sum of forces is equal to zero. • A system is in rotational equilibrium when the sum of the torques is equal to zero. • A system is in static equilibrium when both the sum of forces and the sum of torques are equal to zero. 𝐹=0 𝜏=0 Torque and Static Equilibrium • A child is sitting on a swing. If the child has a mass of 30 kg, what must the forces in the left and right chains be to hold up the child. The swing seat is 0.5 meters long and of negligible mass. Assume the child’s weight is at the center mass of the swing. FL FR R r Fg 𝐹 = 𝐹𝐿 + 𝐹𝑅 − 𝐹𝑔 = 0 𝐹𝐿 + 𝐹𝑅 = 𝐹𝑔 = 𝑚𝑔 = 30 ∗ 10 = 300𝑁 𝜏𝑎𝑏𝑜𝑢𝑡 𝑙𝑒𝑓𝑡 𝑐ℎ𝑎𝑖𝑛 = 0 ∗ 𝐹𝐿 + 𝑅𝐹𝑅 − 𝑟𝐹𝑔 = 0 0 + 0.5𝐹𝑅 − 0.25 ∗ 300 = 0 𝐹𝑅 = 150𝑁 𝐹𝐿 + 150𝑁 = 300 𝑁 𝐹𝐿 = 150𝑁 Torque • What happens when the system is not in static equilibrium? • The same thing that happens for forces! 𝑎 = 𝜏 𝐹 𝑚 = 𝐹 𝑛𝑒𝑡 𝑚 𝜏𝑛𝑒𝑡 𝛼= = 𝐼 𝐼 • Forces cause linear accelerations, torques cause angular accelerations. • The I in the above formula stands for the rotational inertia or moment of inertia and is specific to the object and how the object is rotating. Rotational Inertia or Moment of Inertia • If you remember inertia is an object’s resistance to change. It is directly proportional to mass. • An object’s rotational inertia or moment of inertia is the object’s resistance to rotations. These are impacted by the size, shape and axis of rotation of the object. • I=Ʃmr2 for systems of particles • I=kmr2 for continuous objects, where k is a constant (usually a fraction) • You will NOT need to memorize the moment of inertia for specific objects. Common Moments of Inertia http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html Torque • What is the angular acceleration of a 2 kg disk that has a 5Nm torque applied to it at it’s radius of 0.1m? Idisk= ½ mr2 •𝛼= 𝜏 𝐼 = 𝜏𝑛𝑒𝑡 𝐼 =1 2 5 ∗2∗0.12 = 𝑟𝑎𝑑 500 2 𝑠 Angular Momentum Angular Momentum • • • • Angular momentum is the rotational counterpart of linear momentum. 𝑝 = 𝑚𝑣 𝐿 = 𝑟 × 𝑝 = 𝐼𝜔 There are 2 different representations of angular momentum. One involves the cross product of distance between the axis of rotation and location of object and linear momentum and the other the moment of inertia, I, and angular velocity, ω. • Just as there is a relationship between linear momentum and force, ∆𝑝 = 𝐹∆𝑡, there is also a relationship between angular momentum and torque,∆𝐿 = 𝜏∆𝑡. Conservation of Angular Momentum If no net external torques act on a system then the system’s angular momentum, L, remains constant. Angular Momentum An ice skater begins a spin with his arms out. His angular velocity at the beginning of the spin is 2.0 rad/s and his moment of inertia is 6 kgm2. As the spin proceeds he pulls in his arms decreasing his moment of inertia to 4.5 kgm2. What is the angular velocity after pulling in his arms? Lo L Io I (6)( 2) (4.5) 2.67 rad/s AP Physics C- Rotational Motion- Part II; Dolores Gende