In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may be applied to cause a linear acceleration without an angular acceleration. Calculations in mechanics are often simplified when formulated with respect to the center of mass. It is a hypothetical point where the entire mass of an object may be assumed to be concentrated to visualise its motion. In other words, the center of mass is the particle equivalent of a given object for application of Newton's laws of motion.
In the case of a single rigid body, the center of mass is fixed in relation to the body, and if the body has uniform density, it will be located at the centroid. The center of mass may be located outside the physical body, as is sometimes the case for hollow or open-shaped objects, such as a horseshoe. In the case of a distribution of separate bodies, such as the planets of the Solar System, the center of mass may not correspond to the position of any individual member of the system.
The center of mass is a useful reference point for calculations in mechanics that involve masses distributed in space, such as the linear and angular momentum of planetary bodies and rigid body dynamics. In orbital mechanics, the equations of motion of planets are formulated as point masses located at the centers of mass. The center of mass frame is an inertial frame in which the center of mass of a system is at rest with respect to the origin of the coordinate system.
What is the correct formula for center of mass of a binary system? I had seen that it is
R=\frac{m_{1}r_{1}+m_{2}r_{2}}{m_{1}+m_{2}}
If that is the right equations what exactly does r one and two stand for? I had also seen a much more complicated formula.
Side note: This is not...
Homework Statement
A(n) 64 kg boat that is 8 m in length is initially 8.5 m from the pier. A 32 kg child stands at the end of the boat closest to the pier. The child then notices a turtle on a rock at the far end of the boat and proceeds to walk to the far end of the boat to observe the...
Homework Statement
Let D be the region in the (x, y) plane bounded by the lines y=x, y=4x and the hyperboals xy=1 and xy=9. find the center of mass of D.
Homework Equations
The Attempt at a Solution
My thought: to use the center of mass formula? and use the change of variables?
center of mass and "center of energy"
Hi everyone,
I recently came across a little thought experiment in a book that seemed to imply that the center of mass of an isolated system is not a strictly conserved quantity. Rather, it is the "center of energy" that remains fixed. The idea of a...
How would I find the center of mass (COM) for a ring of variable density, where the variation in density can be described by a continuous mathematical equation.
The density at each location along the ring is described by a function (d) whose independent variables are radius (r) and angle...
Homework Statement
Find the center of mass of the triangle with vertices at (0,0), (6,6) and (-6,6) if the density at (x,y) is equal to y.
Homework Equations
The Attempt at a Solution
I am having trouble with these centroid/center of mass problems, and I can't even figure out the...
1. There are two 50 kg skaters in a friction-less horizontal surface. Both have velocities of |V|=10 m/s, in opossite directions (that is, skater 1's velocity is 10 m/s i, while skater 2's velocity is -10 m/s i). They both skate parallel to each other with a distance of 1,5m. Skater 1 has a...
Homework Statement
A lamina has the shape of the region in the first quadrant that is bounded by the graphs of y = sinx and y= cosx, between x = 0 and x = π/4. Find the centre of mass if the density is δ(x,y) = y.
Homework Equations
I know all the equations for moments and center of...
OMG second time I'm opening a thread in the wrong forum FFS!
Damn bookmarks! MODS move it please.
Homework Statement
Not a home work question, just something i cam across and need a clarification.
Could One show me how to derive to the equation that X(cm)=int(X dm)/M
int=the deformed S of the...
Homework Statement
a solid in the first octant is bounded by the planes y=0 and z=0 and by the surfaces z=4-x^2 and x=y^2. its density function =kxy, k is a constant. find the mass of the solid and the center of mass
Homework Equations
My= double integral of D[x*kxy]dA
Mx=double...
Homework Statement
Given: g=9.8 m/ss
Given: A uniform flexible chain whose mass is 7 kg and length is 5 m.
Given: A small frictionless pulley whose circumference is negligible compared to the length of the chain.
Problem: Initially the chain is hung over the pulley with nearly equal...
To determine the location of the center of mass of a car, the car is driven over a scale. When the front wheels are over the scale, the weight recorded by the scale is 5.70 x 103 N. When the rear wheels are over the scale, the scale reads 6.30 x 103 N. The distance between the front and rear...
Homework Statement
A block of mass M = 15 kg moves with velocity V and explodes at the origin into two
pieces of masses m1 = 10 kg and m2 = 5 kg. The piece m1 moves in the positive direction
of y-axis with speed v1 = 4.5 m/s, whereas the other piece m2 moves in the positive
direction of...
Homework Statement
What is the center of mass a hollow 2D right triangle having legs of length A and B. The sides of the triangle are also extremely thin. Assume uniform density.
Homework Equations
\overline{x}=∫xmdx/∫mdx
The Attempt at a Solution...
A solid hemisphere of radius r given by:
x2+y2+z2= r2, z ≥ 0
Suppose the density of the hemisphere is the constant δ. The coordinates of the center of mass (a,b,c) are given by:
a=(∭ xδdV) / (∭ δdV)
b=(∭ yδdV) / (∭ δdV)
c=(∭ zδdV) / (∭ δdV)
Find a, b ,c
** So far what i...
I tried this problem but only managed to somewhat understand Part A. But that answer is wrong as well, so can someone guide me through steps of finding the answer: Thank you.
To keep the calculations fairly simple, but still reasonable, we shall model a human leg that is 0.92 m long (measured...
Consider a system of several point-masses separated by large distances and moving with relativistic speeds in relation to one another. GR says that gravitational waves travel at the speed of light. SR says that simultaneity often does not exist for such point-masses. Question: Is it possible...
In this problem we will examine the collision between a baseball (the cowhide) and a bat (the ash). We assume a one-dimensional problem. That is, the bat hits the ball squarely, so that the ball reverses its direction after the collision. We also assume that the ball hits the bat at the center...
Homework Statement
I'm building a hovercraft, and I'm familiar with how to calculate center of mass--however, there's something that will be sitting on the surface of the hovercraft that's throwing me off. I just can't seem to get it to work. I'm just trying to find what the mass would be at...
Let vectors vec{A}=(0,0,-3), vec{B}=(2,0,0), and vec{C}=(0,1,1).
Calculate the following, expressing your answers as ordered triples (three comma-separated numbers).
Calculate A x C. For this part, remember that only perpendicular components contribute to the cross-product. The vector C has...
Question:
Determine the location of the center of mass of a "L" whose thin vertical and horizontal members have the same length L and the same mass M. Use the formal definition to find the x and y coordinates, and check your result by doing the calculation with respect to two different...
Homework Statement
A big olive (m = 0.051 kg) lies at the origin of an xy coordinate system, and a big Brazil nut (M = 0.56 kg) lies at the point (0.81, 3.0) m. At t = 0, a force Fo = (2.6 + 1.7) N begins to act on the olive, and a force Fn = (-4.4 -3.4) N begins to act on the nut...
Homework Statement
A baseball player holds a 36-ounce bat (weight = 10.0 Newton) with one hand at the point O. The bat is in equilibrium. The weight of the bat acts along a line 60 centimeters to the right of O. What is the torque exerted on the center of mass of the bat by the player...
1. Ball 1, ball 2, and ball 3 have masses 1 kg, 2 kg, and 3 kg respectively, and are initially arranged at x = 5 m, x = 10 m, and x = 15 m respectively (see picture). y = 5 m for all three balls. Now suppose that a net force of 1 N is applied to ball 1 in the +y direction and a net force of 2 N...
1. Ball 1, ball 2, and ball 3 have masses 1 kg, 2 kg, and 3 kg respectively, and are initially arranged at x = 5 m, x = 10 m, and x = 15 m respectively (see picture). y = 5 m for all three balls. Now suppose that a net force of 1 N is applied to ball 1 in the +y direction and a net force of 2 N...
Homework Statement
Consider the Sun and the Earth as a 'system' of two 'particles'. The center of mass of this system lies a distance L from the center of the Sun. Find L and compare it to the radius of the Sun.
The center of mass of the Earth-Sun system lies?
A. inside the radius of the...
Homework Statement
A man decides to boat out on the lake for a relaxing day of fishing. At one point during the day, he finds himself facing the shore. He judges that he is a distance Dinit = 35 ft from the shore. The man has a mass of Mman = 190 lbs, and the boat a mass of Mboat = 270 lbs. The...
Homework Statement
A puck with a mass of 0.08 kg and a radius of 0.04 m slides along an air table at a speed of 1.5 m/s. It makes a glancing collision with a second puck at rest having a radius of 0.06 m and a mass of 0.12 kg such that their rims just touch. Because their rims are coated...
Homework Statement
In Fig. 9-39, three uniform thin rods, each of length L = 28 cm, form an inverted U. The vertical rods each have a mass of 12 g; the horizontal rod has a mass of 31 g. What are (a) the x coordinate and (b) the y coordinate of the system's center of mass? (Give your answer in...
Homework Statement
alley club-ax consists of a symmetrical 8 kg stone attached to the end of a uniform 2.5 kg stick that is 98 cm long. How far is the center of mass from the angle end of the club-ax?
Homework Equations
Xcm = (sum) x*m / m
The Attempt at a Solution
I don't know...
Is it because less force (from, say, potential objects that could tip the object) is required to exert the same amount of torque on objects with higher center of mass?
1. A pellet gun fires ten 2.0 g pellets per second with a speed of 500 m/s. The pellets are stopped by a rigid wall. What are (a) the magnitude of the momentum of each pellet, (b) the kinetic energy of each pellet, and (c) the magnitude of the average force on the wall from the stream of...
Homework Statement
If x is the height of the remaining soda at any time in a can of soda, find x (in terems of M, H, and m) when the center of mass is the lowest. {{The soda is draining out of the bottom of the can}} Also, the Can and soda have uniform density.
So the can is set up so...
Homework Statement
A rod of length 24.5 cm has linear density (mass-per-length) given by the following equation, where x is the distance from one end.
λ = 50.0 g/m + 20.5x g/m2
(a) What is its mass?
(b) How far from the x = 0 end is its center of mass?
Homework Equations...
Urgent ! Center of mass and relevant stuffs
Homework Statement
A vertical spring of constant k is attached at the bottom to a platform of mass mp, and at the top to a massless cup. The platform rests on a scale. A ball of mass mb is placed in the cup. What is the reading on the sacle when...
Homework Statement
A baseball bat of length L has a peculiar linear density given by \lambda = \lambda0 * (1+X2/L2)
Find the x coordinate of the center of mass in terms of L
Homework Equations
Mxcm= mr
The Attempt at a Solution
So I use integration
The integrand I have is x*\lambda dx...
Homework Statement
A box with its center of mass off-center as
indicated by the dot is placed on an inclined
plane.
In which orientation does the box tip over?
Homework Equations
Teacher didn't even mention anything related to this
The Attempt at a Solution
Since the CM was at...
Hey guys, does anyone know a good website teaching center of mass? I need to know how to find the center of mass of objects like spheres, cubes etc. and also do problems such as "a man is standing on a boat and moves from one side to the other, how far does the boat move." Stuff like that...
Homework Statement
To determine the location of the center of mass of a car, the car is driven over a scale. When the front wheels are over the scale, the weight recorded by the scale is 5.05 x 103 N. When the rear wheels are over the scale, the scale reads 6.20 x 103 N. The distance between...
1. A right circulat cone of constant density (5kg per m^3) is 4 meters from the base to the tip. the diameter of the base is 6 m. find each of the following using inergrals
A) Find the volume and mass of the cone
B) find the center of mass of the cone
C) find the moment of inertia of the...
Homework Statement
From an inertial reference frame S, the vector position of a particle of mass
m1 = 1kg is given by r1(t)=(txhat - t^2yhat)m.
The vector position of a particle m2=2m1 is given by r2=(t)=(txhat +t^3yhat)m
Assume t=1second
Find the position, the velocity and the...
If you guys would be kind enough to show me if my attempt to problems posted are correct or incorrect.
Homework Statement
A uniform 3.0 kg cylinder of radius 0.15 m is suspended by two strings wrapped around it, as shown in the figure below. As the cylinder descends, the strings unwind from...
Homework Statement
I am very fuzzy about how to find the center of mass of an object and, similarly, the moment of inertia for an object using integrals. Here are the formulas as I remember them.
Center of mass: (1/M) \int \vec{r} dm
Moment of Inertia: \int R^{2} dm
What I was...
A uniform spherical shell of mass 9.00 with diameter 54.0 has four small masses of mass 2.00 attached to its outer surface and equally spaced around it. This combination is spinning about an axis running through the center of the sphere and two of the small masses
What friction torque is...
Homework Statement
Find the mass and center of mass of the solid bounded by the planes x=0, y=0, z=0, x+y+z=1; density\delta(x,y,z)=y
Homework Equations
M=\int\int_D\int\delta dV
M_{yz}\int\int_D\int x \delta dV;M_{xz}\int\int_D\int y \delta dV;M_{xy}\int\int_D\int z \delta dV...
Before begining, I would like to point out I did look at the following thread previous to this one, and thus have a few questions about that thread and it's application to this one: https://www.physicsforums.com/showthread.php?t=143536
Homework Statement
The problem...
Homework Statement
A pendulum consists of a uniform disk with radius r=0.100m and mass 0.500 kg attached to the end of a uniform rod with length L=0.500 m and m 0.250 kg. It pivots at the other end of the rod. a) Calculate the rotational inertia of the pendulum about the pivot point. b) What...
Homework Statement
3 identical rectangular blocks on frictionless surface have equal magnitudes of force and in the same direction are exerted on them. The CoM is the same for all the blocks and in the center of the rectangle. For block 1, the force is exerted to the left of the CoM, and...
Homework Statement
http://img201.imageshack.us/img201/2135/21592808vx2.th.jpg http://g.imageshack.us/thpix.php
In class you learned that when a person walks, their centre of mass must be positioned directly above the foot their weight is resting on. The figure shows the anatomy of the...