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## Chapter 4: Rigid bodies. (III) – Problems

ROTATIONAL DYNAMICS: EXERCISES

1. A mass M = 120 kg hangs from a beam which is held by a cable.
1. What is the tension T in the cable?
2. Does the wall exert any force on the beam? Which one?
2. A stone of mass 0.6 kg is forced to travel in a circle while it hangs from the ceiling, as shown in Figure. If the angle between the rope and the vertical is θ = 30º and the radius of the circle is r = 35 cm:
1. What is the speed of the stone?
2. What is the tension in the rope?
3. A wheel of radius r = 80 cm has moment of inertia 10 kg m2. It is rotating around its central axis propelled by a rocket attached to a point on its outer rim. The rocket is expelling gas
tangentially to the wheel, resulting in a constant force. Determine:

1. The magnitude of the equivalent force, if we know that the wheel, starting from rest, reaches an angular speed of 1 rev/s in 6 s.
2. The value of both tangential and normal acceleration in a point on the outer rim of the wheel.
3. The angle that the total acceleration forms with the radius at that point.
4. The time that the wheel takes to reach the same angular velocity, under the action of the same force, if we add a very thin ring of mass 5 kg around the outer rim.

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## Chapter 4: Rigid bodies (III)

Chapter 4: RIGID BODIES III

1. ROTATION. CIRCULAR MOTION.
2. MOMENT OF A FORCE (TORQUE).
3. MOMENT ANGULAR CONSERVATION.

Massachusetts Institute of Technology have developed Open Course Wares (OCW) where you can follow a “Classical Mechanics course” by Internet.

One of the most important magnitudes in architecture is the moment of a force or torque. And this magnitude will be extremely important in Chapter 5: Equilibrium.

Linear momentum, i.e. p = m·v, has a rotational analogue that it is called angular momentum, L. For a symmetrical object rotating about a fixed axis through the centre of mass (CM), the angular momentum is L = I·ω where I is the moment of inertia and ω is the angular velocity about the axis of rotation. The SI units for L are kg·m^2/s, which has no special name.

We saw in Chapter 2: Dynamics laws and applications that Newton’s second law can be written more generally in terms of momentum ΣF = Δp/Δt. In a similar way, the rotational equivalent of Newton’s second law which is Στ = I·α, can also be written more generally in terms of angular momentum Στ = ΔL/Δt where Στ is the net torque acting to rotate the object and ΔL is the change in angular momentum in a interval time Δt.

Angular momentum is an important concept in physics because, under certain conditions, it is a conserved quantity. If Στ = ΔL/Δt on an object is zero then ΔL = 0, so L does not change. This is the law of conservation of angular momentum for a rotating object:

The total angular momentum of a rotating object remains constant if the net torque acting on it is zero.

• A video tutorial about rotation can be watched on the MIT-OCW webpage:

https://ocw.mit.edu/courses/physics/8-01sc-classical-mechanics-fall-2016/week-10-rotational-motion/30.1-introduction-to-torque-and-rotational-dynamics

• A video tutorial about the moment of a force/torque can be watched on the MIT-OCW webpage:

https://ocw.mit.edu/courses/physics/8-01sc-classical-mechanics-fall-2016/week-10-rotational-motion/30.4-torque

• A video tutorial about rotational dynamics can be watched on the MIT-OCW webpage:

https://ocw.mit.edu/courses/physics/8-01sc-classical-mechanics-fall-2016/week-10-rotational-motion/31.1-relationship-between-torque-and-angular-acceleration

• A video tutorial about solved problems can be watched on the MIT-OCW webpages:

Worked example: moment-of-inertia-of-a-disc-from-a-falling-mass

Worked example: massive-pulley-problems

MisConceptual Questions

1. The symmetric simple truss is loaded as shown in Figure. Which force shown exerts the largest magnitude torque on the truss around point A? And around point B?
2. Calculate the net torque around point O due to the forces acting on the plate shown.

Video lecture: Rotational dynamic example

Problem set number 5

1. A wheel of radius r = 80 cm has moment of inertia 10 kg·m^2. It is rotating around its central axis propelled by a rocket attached to a point on its outer rim. The rocket is expelling gas
tangentially to the wheel, resulting in a constant force. Determine:

1. The magnitude of the equivalent force, if we know that the wheel, starting from rest, reaches an angular speed of 1 rev/s in 6 s.
2. The value of both tangential and normal acceleration in a point on the outer rim of the wheel.
3. The angle that the total acceleration forms with the radius at that point.
4. The time that the wheel takes to reach the same angular velocity, under the action of the same force, if we add a very thin ring of mass 5 kg around the outer rim.

If you have some doubts, you can watch next video related to rotational dynamic (Professor Michel van Biezen):

If you were satisfied with this example, you can check more video lectures on the webpage:

http://ilectureonline.com