Rigid Body Dynamics
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Equilibrium of Rigid Body
In this section, we will discuss the static equilibrium of rigid bodies.A rigid body is said to be in static equilibrium when the resultant force acting on the body is zero and the resultant torsion at any point of the rigid body is zero.Mathematically, the above statement is written as,
ΣF = 0 or ΣFx = 0 and ΣFy = 0 ...... 1
and
Σ 𝞃 = 0 ........................ 2
Equation 1 is called the first condition for equilibrium and equation 2 is the second condition for equilibrium.
If the first condition for equilibrium, ΣF= 0, is fulfilled, the body is in translational equilibrium. A rigid body is said to be in a complete equilibrium when both conditions for equilibrium are fulfilled. Static equilibrium can be classified into three, i.e. stable equilibrium, unstable equilibrium, and indifferent (neutral) equilibrium.
a. 2 cm
b. 4 cm
c. 6 cm
d. 8 cm
e. 10 cm
2) A solid cylinder made of iron is rolling down the floor at 10 m/s velocity. Its mass and diameter is 4 kg and 80 cm respectively. The total kinetic energy of the cylinder is ...........
a. 250 Joule
b. 300 Joule
c. 400 Joule
d. 500 Joule
e. 625 Joule
3) The rod AC has the mass of 40 kg and the length of 3 m. The footing distance of A and B is 2 m (the block can actually rotate at B). A boy with mass of 25 kg mass walks from A to C. The minimum distance from C for the boy so that the block would still be in balance (with the tip of A slightly up) is ........
a. zero
b. 0.1-meter
c. 0.2-meter
d. 0.3-meter
e. 0.4 meter
4) For an object in unstable equilibrium, after a disturbance is given, the position of the center of gravity will be ..........
a.. the same
b. higher
c. lower
d. up-side-down
e. higher and lower
5) A solid sphere with the diameter of 20 cm rotates with the axis at the center of the sphere. The sphere has the angular velocity of ω = (10 + 25t) rad/s, where t is in seconds. If the mass of the sphere is 4 kg, the magnitude of the torsion working on the sphere is ...........
a. 0.32 N.m
b. 0.4 N.m
c. 0.65 N.m
d. 0.8 N.m
e. 1.6 N.m
6) A grindstone has 4 kg mass and 10 cm radius. If a fixed moment of force is applied, the grindstone will reach an angular velocity of 1200 rpm in 20 s. If the grindstone is initially at rest and having shapes of a solid cylinder, the magnitude of its moment of force is ..... N.m
a. 8 π x 10-2
1. Stable Equilibrium
A marble has the center of gravity at the center of the sphere. If the marble is put in a hemispherical (concave) container, it will stay at rest (be in an equilibrium state) at the bottom of the container. If the marble is given a disturbance by pushing it, its center of gravity will raise. It is indicated by the raise of the position of the marble in the container.If the disturbance is removed, the marble will return to its initial equilibrium. This kind of equilibrium is called stable equilibrium. A stable equilibrium is characterized by the raise of the center of gravity of the object whenever a disturbance is given. When the disturbance is removed, the object will return to its previous position. A rocking chair is another example of an object in a stable equilibrium.
2. Unstable Equilibrium
If a marble is so carefully put on top of a ball that it reaches equilibrium (still) and then given disturbance, the marble will never return to its initial position. This kind of equilibrium is known as the unstable equilibrium. It is characterized by the lowering of object's center of gravity whenever return to its initial position. An upright piece of wood is one example of unstable equilibrium
3. Indifferent (Neutral) Equilibrium
Now, consider a marble put on a flat and slippery floor. Whenever the marble is given a disturbance, its center of gravity will not undergo any change in height. When the disturbance is removed, the marble will soon be in its new equilibrium position. This is known as indifferent or neutral equilibrium.
It is characterized by the unaffected position of the center of gravity before and after a disturbance is given. Another example of indifferent equilibrium is cylinder put on a plane floor.
B. Center of Gravity of an Object
An object can be considered a composite of many particles. Each particle has weight. The weight of an object is the resultant of all gravitational forces undergone by the particles that make up the object. The direction of the gravitational force each particle is toward the center of the earth.
The resultant of all gravitational forces (weight forces) of the particles that make up an object is located at a certain point. The capture point of the gravitational force is called the center of gravity of the center of mass of an object. What about the center of gravity of two or three-dimensional objects? Objects that have symmetries such as a triangle plate, polygon, round plate, cylinder, and ball are shaped regularly and are homogeneous. Hence the weight center is at their respective equilibrium point.
Exercise
- Three particles with masses of 2 kg, 4 kg, and 6 kg respectively are located at the three ends of a triangle of 1.5 m length on its sides.
- On a rod AB, a block with masses of 8 kg is attached. At a distance of 1 m from point A, a load x with masses of 6 kg is placed. If the length of the rod AB is 5 m, calculate the tension of the rope T.
- Why is the final velocity of an object sliding on an inclined plane faster than an object rolling on it?
- In a wheel of 12 kg.m2 moments of inertia, a constant torsion of 50 N.m is applied. Determine the angular acceleration.
- A solid sphere rolls from its original position down an inclined plane with the height of 1.4 m. Determine the linear velocity of the sphere at the bottom of the inclined plane. ( g = 10 m/s2)
- A particle with masses of 1 g rotates around an axis at 90 rotations per minute. If the distance between the particle and the axis is 50 cm, calculate its angular momentum.
Multiple Choice
1) A particle revolves with angular velocity of 10 rad/s. If the mass of the particle is 2 g and the angular momentum is 8 x 10-6 kg m2/s, the trajectory radius will be ........a. 2 cm
b. 4 cm
c. 6 cm
d. 8 cm
e. 10 cm
2) A solid cylinder made of iron is rolling down the floor at 10 m/s velocity. Its mass and diameter is 4 kg and 80 cm respectively. The total kinetic energy of the cylinder is ...........
a. 250 Joule
b. 300 Joule
c. 400 Joule
d. 500 Joule
e. 625 Joule
3) The rod AC has the mass of 40 kg and the length of 3 m. The footing distance of A and B is 2 m (the block can actually rotate at B). A boy with mass of 25 kg mass walks from A to C. The minimum distance from C for the boy so that the block would still be in balance (with the tip of A slightly up) is ........
a. zero
b. 0.1-meter
c. 0.2-meter
d. 0.3-meter
e. 0.4 meter
4) For an object in unstable equilibrium, after a disturbance is given, the position of the center of gravity will be ..........
a.. the same
b. higher
c. lower
d. up-side-down
e. higher and lower
5) A solid sphere with the diameter of 20 cm rotates with the axis at the center of the sphere. The sphere has the angular velocity of ω = (10 + 25t) rad/s, where t is in seconds. If the mass of the sphere is 4 kg, the magnitude of the torsion working on the sphere is ...........
a. 0.32 N.m
b. 0.4 N.m
c. 0.65 N.m
d. 0.8 N.m
e. 1.6 N.m
6) A grindstone has 4 kg mass and 10 cm radius. If a fixed moment of force is applied, the grindstone will reach an angular velocity of 1200 rpm in 20 s. If the grindstone is initially at rest and having shapes of a solid cylinder, the magnitude of its moment of force is ..... N.m
a. 8 π x 10-2
b. 4 π x 10-2
c. 2 π x 10-2
d. 8 π x 10-4
e. 4 π x 10-4
7) A solid sphere undergoes translation and rotation with linear and angular velocities v and ω respectively. The total kinetic energy of the sphere is ..........
c. 2 π x 10-2
d. 8 π x 10-4
e. 4 π x 10-4
7) A solid sphere undergoes translation and rotation with linear and angular velocities v and ω respectively. The total kinetic energy of the sphere is ..........
a. 6/10
mv2
b. 7/10 mv2
c. 8/10 mv2
d. 9/10 mv2
e. 11/10 mv2
You Need To Know
Have your ever seen an acrobatic performance? Walking on a rope is one of them. How Could the man possibly keep his balance on a rope?
By using his hands, the acrobatic player controls his steps in such a way that the resultant of the moment of force is zero, as required to reach equilibrium state of a rigid body. If the rope is considerably long, the players will spread his arms to reduce his weight. If the rope is somewhat short, the arms do not have to be spread.
Case Study
The design of racing cars is much more stable than the design for trucks. That is because the racing cars have lowered center of gravity and their base is also wider. That kind of design prevents the racing cars from rolling up-side-down while taking on curves with high speed. Whereas trucks or buses will easily roll over whenever they take on curves with high speed.
Why does the object with the higher center of gravity tend to be a more unstable tan object with the lower center of gravity, despite having the same base?
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