Mechanics 5 Dynamics of a rigid body Torque Moment of Inertia Newton s laws for a rigid body Angular momentum Conservation law Basic phenomena In an empty space with no external forces acting on the body, it is impossible to change the velocity of a particle. However it is possible to change the rotational frequency of a body using only internal forces. This is done by changing the mass distribution around the rotation axis. If no external forces exists, the centre of mass of the system of particles stays at rest of continues with a constant velocity. Centre of mass 1
Experiment 1 A B 1 kg 1 kg Two identical, massive wheels are fixed to the wall. The wheels consist of two disks with different radii. In the wheel A the rope has been wrapped around the larger disk and in B around the smaller disk. In both A and B there is a 1 kg mass hanging from the ropes. Question: Which of the masses falls with the greatest acceleration? Answer: Though the forces are equal, A falls faster, because the point of action is further away from the axis of rotation. The force has a greater torque on the wheel. Torque T The rotating effect of a force depends not only on the magnitude of the force F, but also on the distance of the action line of the force from the rotation axis. Definition: The torque of force F with respect to axis point A is defined by T = F r where r is the distance of the action line of the force from the point A The unit of Torque is 1 Nm (Newton meter) r F T = Fr 2
Experiment 2 A = a hollow cylinder, B) a solid cylinder C) a solid ball All have the same mass and radius A B C 1 kg 1 kg 1 kg Question: In what order the masses fall down? Answer: The mass C is first down B is second and A is third. Argumentation: Even though the masses are same, the distribution of mass around the axis varies. In the ball the mass is distributed closest to the axis. That s why the ball has the least inertia and it falls down fastest. In the empty cylinder A the all the mass is at the distance r from the axis and that is why it is the most difficult to get into a rotation. Rotational kinetic energy r C v c C B A rotating body has kinetic energy, which is the sum of the kinetic energies of its mass points. E rot = S½ m i v i 2 A r A Because all the mass points have different velocities, but same angular velocities, it is more convenient to write v i = wr i. Then E rot = ½(S m i r i2 )w 2 E rot = ½ I w 2 Quantity I =S m i r i2 or I =Ûr 2 dm is called the moment of inertia of the body. It describes the distribution of mass around the axis. 3
Moment of Inertia The moment of inertia of a rigid body is defined by I = Ú r 2 dm where integration goes through all the mass elements of the body and r is the distance of mass element dm from the axis. Table of moments of inertia of most common bodies: Hollow cylinder (mass m, radius r) I = mr 2 Solid cylinder I = ½ mr 2 Solid ball I = 2/5 mr 2 Stick (mass m, length l ) - axis = midpoint I = 1/12 ml 2 - axis = end of the stick I = 1/3 m l 2 Steiner s rule Let I 0 = the moment of inertia of a body with respect to an axis A, which goes through the centre of mass of the body. Then the moment of inertia with respect to any axis A parallel to A can be calculated from I = I 2 0 + m a a = the perpendicular distance of A and A 4
Analogy between linear motion and rotation The formulas of linear and rotational motion are analogous. You have only to know what quantities in the linear motion and in the rotational motion correspond each other. Below is a table of corresponding quantities distance s velocity v = Ds/ D t acceleration a = D v/ D t mass m force F linear momentum p = mv angle j angular velocity w = Dj/ D t angular acceleration a = D w / D t the moment of inertia I torque T angular momentum L = I w Analogous formulas linear kinematics: rotation: v = v 0 + at w = w 0 + w t s = v k t = v 0 t + ½ at 2 j = w k t = w 0 t + ½a t 2 dynamics: dynamics of rotation: Newton s II law: F = ma T = I a work: W = F s W = T j power: P = F v P = T w kinetic energy: E k = ½ mv 2 rotational energy: E rot = ½ I w 2 The conservation law of linear momentum -> The conservation law of angular momentum 5
Example1 A wheel (a solid cylinder) (m=5.0 kg, r = 30 cm, n = 900 RPM) is stopped by using a breaking force of 20 N. Calculate a) the angular retardation a b) in how many seconds does the wheel stop c) how many rounds does the wheel rotate before stopping Solution: From T = I Dw / Dt we get Dt = I Dw / T = ½ mr 2 Dw / Fr Now Df = 900 RPM = 900/60 RPS = 15 Hz => Dw = 2p f = 94.2 rad/s breaking time: D t = ½*5*0.3*94.2 / 20 = 3.5 s Number of rounds = average frequency * time = 15/2 round/s*3.5 s = 26.5 rounds Part C could be solved also using energy principle During the breaking the rotational energy transforms to the work done by the friction ½ I w 02 = T j => j = ½ I w 0 2 / T = ½ ( ½ mr 2 w 02 /Fr) = ¼ mr w 02 /F = ¼ 5*0.3*94.24 2 /20 rad = 166 rad In rounds: 166/2π = 26 rounds 6
Pure rolling = rolling without gliding r P w speed v road In pure rolling: The radial velocity of point P with respect to the center v = w r ( rolling condition ) v is also the linear speed of the wheel Example: A wheel has a radius of 50 cm and it rolls with 3 RPS. The its speed is r w = 0.5 m * 2p*3 1/s = 9.4 m/s Example 2 competition between shapes a b c h A hollow cylinder, a solid cylinder and a solid ball start from rest rolling down the hill, with height difference h = 3.0 m. a) In what order do they come down? b) Calculate their final velocities. Solution: a) The order is ball, solid cylinder, hollow cylinder. The ball is first (smallest moment of inertia), the solid cylinder is second because its moment of inertia is next to the ball. 7
cont b) We use the formulas from the table of moments of inertia: hollow cylinder I = mr 2, solid cylinder I = ½ mr 2, solid ball I = 2/5 mr 2. rolling condition when a round body (radius r, angular velocity w, speed v) rolls without gliding, the velocity of a point on the radius with respect to the centre is equal to the linear velocity of the body: v = w r Energy principle: The potential energy the body has on the top transforms partly to kinetic energy, partly to rotational energy mgh = ½ mv 2 + ½ I w 2 cont Replacing these conditions, we get for the hollow cylinder: mgh = ½ mv 2 + ½ Iw 2 = ½ mv 2 + ½ (mr 2 ) v 2 /r 2 = ½ mv 2 +½ mv 2 = mv 2 => v = (gh) = (9.81*3.0) m/s = 5.4 m/s and for the solid cylinder mgh= ½ mv 2 + ½ Iw 2 = ½ mv 2 + ½ (½ mr 2 ) v 2 /r 2 = ½ mv 2 + ¼ mv 2 = 3/4 mv 2 => v = (4/3gh) = (4/3*9.81*3.0) m/s = 6.3 m/s and for the ball mgh=½mv 2 + ½ I w 2 = ½ mv 2 + ½ (2/5 mr 2 ) v 2 /r 2 = ½ mv 2 + 2/5 mv 2 =7/10 mv 2 josta v = (10/7gh) = (10/7*9.81*3.0) m/s = 6.5 m/s 8
Angular Momentum (spin) L = Iw L The angular momentum is a vector in the direction of rotational axis and magnitude of L = I w Conservation law: The angular momentum of an isolated system is a constant * Angular momentum is proportional to the rotational frequency and the mass of the rotating body. Also the mass distribution around the axis influences it through the moment of inertia. T, w and L as vectors In picture on the right the right hand thumb shows the direction of the angular momentum vector w and spin vector L = I w Also the torque is defined as vector T = r x F r F T axis point Counterclockwise force has a Torque, that is directed to us (the red arrow) 9
The conservation of angular momentum The angular momentum of an isolated system is a constant means that - Rotating bodies tend to preserve the direction of the rotation axis, and the rotational frequency - If the moment of inertia of a body increases for some reason, it s rotational frequency must decrease Applications of the conservation law 1. Flywheels in motors: Many motors have a heavy flywheel which keep motors going steady 2. Riffling of a gun: The pipe of a gun has riffling in order to force the bullet into a rotational motion. According to the conservation law the bullet keeps its axis direction during the flight. 3. Tail rotors in helicopters prevent them from rotating horizontally. a wheel of Volkswagen tail rotor of Sikorsky helicopter 10
Spinning top A rotating body with a big moment of inertia is called a spinning top A spinning top tends to keep it s axis of rotation More applications In ships and tanks there are heavy stabilizing wheels The aero planes have rotating wheels showing the horizon A figure skater spins fast by decreasing her moment of inertia and in that way increasing her rotational frequency Gymnastic can also control his rotational frequency by changing his mass distribution around the axis figure skating artificial horizon of an aircraft 11
Gyroscopic stabilizers of USS Henderson, (battleship built 1917) A photographer has a camera with gyroscopic stabilizer, when he takes pictures from a helicopter Gyroscopes The gyroscope effect was discovered in 1817 by Johann Bohnenberger and invented and named in 1852 by Léon Foucault for an experiment involving the rotation of the Earth. A gyroscope is a device for measuring or maintaining orientation, based on the principle of conservation of angular momentum. In physics this is also known as gyroscopic inertia or rigidity in space. Gyroscopes are used in autopilots of aeroplanes. Gyroscope maintains its spatial directions despite the rotation of the Earth 12
Gyrocompass Gyrocompass consists of a rotating wheel, which is fixed from both ends of its axis to a plate, which can turn freely in a horizontal plane. The wheel axis of gyrocompass turns to the North. The phenomenon is not based on magnetism, and thus not effected by magnetic disturbances or presence of iron. Gyrocompass is used in ships and planes. A human gyrocompass Gyrocompass of an aircraft west Principle of the gyrocompass The Earth east The rotation of Earth would causes a torque to the wheel. The direction of the torque is to the North. According to the law T dt = DL also the spin turns to the North. When spin point to the North, the wheel doesn t experience torque anymore. 13
Impulse principle Newton s II law for rotation: T = I a T = I Dw/Dt T Dt = I DL This means that Torque vector is always in the direction of the change of angular momentum DL (and Dw). The spin chases the torque This leads to very non-intuitive phenomena. In the next section some of them are explained. Torque Precession of a spinning top P r Spin Question: What happens if a spinning top rotates in non-vertical position? Gravitation Answer: The torque vector due to gravitation points out from the paper (towards us). The spin vector chased the torque and starts slowly to rotate. This is called precession The Earth s axis rotates with a period of 26000 years. This was first observed by a Greek astronomer Hipparchus. From this follows that the Arctic Circle moves every year a few centimeters. 14
Precession of a wheel r F rope A spinning disk is hanging from a rope, which is fixed to the other end of the axis. What happens? L - vector Torque - vector Force F=mg Answer: The wheel axes starts to rotate slowly to the right maintaining its horizontal position. The spin L chases the torque T Coriolis force If a stone is dropped from a helicopter above Rovaniemi, is doesn t hit the ground at Rovaniemi, but to the South of Rovaniemi. This is seen the picture: Rovaniemi moves along the red line and the particle moves along the light blue line around the Earth This phenomenon is called the Coriolis force. (Of course there is no such force. The reason is the rotation of the Earth and the law of inertia: (Newton s I law). In the southern hemisphere the stone would hit the Earth to the North of the place where it is dropped. 15
Meteorological consequence In the Northern hemisphere the Coriolis force deflects the wind to the right and makes the air move counterclockwise around the centre of the low pressure system. In the Southern hemisphere it is just the opposite. Kepler s III law conservation of spin A2 planet A1 sun The law of equal areas says that the radius from the Sun to the planet sweep equal areas in equal times. Mathematically A = ½ r 2 Dj : => ½ r 12 Dj 1 = ½ r 12 Dj 1 m r 12 Dj 1 / Dt = m r 12 Dj 1 / Dt I 1 w 1 = I 2 w 2 L 1 = L 2 Kepler found the conservation law of spin already in 16 th century in this special case, just analyzing observation. When the planet is far from the Sun, its moment of inertia I decreases. That s why its angular velocity w increases so that the product I w remains the same. 16