In: Physics

The three masses shown in (Figure 1) are connected by massless, rigid rods. Assume that m_{1} = 180 g and m_{2} = 350 g.

What is the x-coordinate of the center of mass?

Express your answer with the appropriate units.

What is the y-coordinate of the center of mass?

Express your answer with the appropriate units.

Concepts and reason

The main concepts required to solve this problem are the center of mass and distance. Initially, write the equations for the \(x\) and \(y\) -coordinates of the center of mass of the system of three masses. Use this equation and calculate the x-coordinate of the center of mass and y-coordinate of the center of mass of the three masses system.

Fundamentals

Center of mass can be defined as the point in which all the particles of the masses of the system is supposed to be concentrated. The x-coordinate of the center of mass of the system of three masses is, \(x_{C M}=\frac{m_{1} x_{1}+m_{2} x_{2}+m_{3} x_{3}}{m_{1}+m_{2}+m_{3}}\)

Here, \(m_{1}\) is the first mass, \(m_{2}\) is the second mass, \(m_{3}\) is the third mass, \(x_{1}, x_{2},\) and \(x_{3}\) are distances of the three masses from the origin respectively in x-axis. The y-coordinate of the center of mass of the system of three masses is, \(y_{C M}=\frac{m 1 y 1+m 2 y 2+m 3 y 3}{m_{1}+m_{2}+m_{3}}\)

Here, \(y_{1}, y_{2}\) and \(y_{3}\) are the vertical distances of the three masses from the origin respectively.

(A) The equation for the x-coordinate of the center of mass of the three masses system is, \(x_{C M}=\frac{m_{1} x_{1}+m_{2} x_{2}+m_{3} x_{3}}{m_{1}+m_{2}+m_{3}}\)

Substitute \(180 \mathrm{~g}\) for \(m_{1}, 12 \mathrm{~cm}\) for \(x_{1}, 350 \mathrm{~g}\) for \(m_{2}, 12 \mathrm{~cm}\) for \(x_{2}, 200 \mathrm{~g}\) for \(m_{3},\) and 0 for \(x_{3}\) in above equation.

$$ \begin{array}{c} x_{C M}=\frac{(180 \mathrm{~g})(12 \mathrm{~cm})+(350 \mathrm{~g})(12 \mathrm{~cm})+(200 \mathrm{~g})(0)}{(180 \mathrm{~g})+(350 \mathrm{~g})+(200 \mathrm{~g})} \\ =8.71 \mathrm{~cm} \end{array} $$

Part A The x-coordinate of the center of mass of the three masses is \(8.71 \mathrm{~cm} .\)

Explanation \(\mid\) Common mistakes | Hint for next step

Here, the x-coordinate of the center of mass of the three masses is depending on their masses and their distances from the origin in x-axis. The horizontal distance of the mass \(\mathrm{m} 3\) is 0 because the third mass is located at the origin.

(B) The equation for the y-coordinate of the center of mass of the three masses is, \(y_{C M}=\frac{m_{1} y_{1}+m_{2} y_{2}+m_{3} y_{3}}{m_{1}+m_{2}+m_{3}}\)

Substitute \(180 \mathrm{~g}\) for \(m_{1}, 0\) for \(y_{1}, 350 \mathrm{~g}\) for \(m_{2}, 10 \mathrm{~cm}\) for \(y_{2}, 200 \mathrm{~g}\) for \(m_{3},\) and 0 for \(y_{3}\) in above equation.

$$ \begin{array}{c} y_{C M}=\frac{(180 \mathrm{~g})(0)+(350 \mathrm{~g})(10 \mathrm{~cm})+(200 \mathrm{~g})(0)}{(180 \mathrm{~g})+(350 \mathrm{~g})+(200 \mathrm{~g})} \\ =4.79 \mathrm{~cm} \end{array} $$

Part B The y-coordinate of the center of mass of the three masses is \(4.79 \mathrm{~cm}\).

Explanation | Common mistakes

Here, the y-coordinate of the center of mass of the three masses is depending on their masses and their distances from the origin in y-axis.

Part A

The x-coordinate of the center of mass of the three masses is \(8.71 \mathrm{~cm}\)

Part B

The y-coordinate of the center of mass of the three masses is \(4.79 \mathrm{~cm}\).

Air flows through the tube shown in the figure. Assume that air is an ideal fluid.
What is the air speed v1 at point 1?
What is the air speed v2 at point 2?
What is the volume flow rate?

The current in the wire shown in (Figure 1) is increasing.
Part A
What is the direction of the induced current in loop A?
clockwise
counterclockwise
No current is induced in this loop.
Part B
What is the direction of the induced current in loop B? clockwise
counterclockwise
No current is induced in this loop.

What is the electric potential energy of the group of charges in (Figure 1)? Assume that q = -5.5 nC.
U= (value) (units)

A copper loop hangs from two strings, as shown in (Figure 1).
Part B
In which direction (toward or away from the magnet) does the loop swing if the magnet is moving toward the loop?
toward the magnet
away from the magnet Figure
The loop doesn't move
Part C
In which direction (toward or away from the magnet) does the loop swing if the magnet is moving away from the loop?
toward the magnet
away from the magnet
The loop...

Consider the circuit shown in (Figure 1). Suppose that E = 15 V . include units with answers.
Part A: Find the current through the resistor a.
Part B: Find the potential difference across the resistor a. answer: 7.5 V
Part C: Find the current through the resistor b.
Part D: Find the potential difference across the resistor b.
Part E: Find the current through the resistor c.
Part F: Find the potential difference across the resistor c.
Part G: Find...

The container shown in (figure 1) is filled with oil. It is open to the atmosphere on the left.
What is the pressure at point A?
Pa= kPa
What is the pressure difference between points A and B?
Pb-Pa= kPa
What is the pressure difference between points A and C?
Pc-Pa= kPa

The four 1.0 g spheres shown in the figure are released simultaneously and allowed to move away from each other.
What is the speed of each sphere when they are very far apart?

if f and g are the functions whose graphs are shown, let u(x) = f(x)g(x) and v(x) = f(x)/g(x)
(a) Find u'(1)
(b) Find v'(5).

Rigid bar ABC shown in Figure P1.30 is supported by a pin at bracket A and by tie rod (1). Tie rod (1) has a diameter of 5 mm, and it is supported by double-shear pin connections at B and D. The pin at bracket A is a single-shear connection. All pins are 7 mm in diameter. Assume a = 600 mm, b = 300 mm, h = 450 mm, P = 900 N, and θ = 55°. Determine the...

When a magnet is plunged into a coil at speed v, as shown in the figure, a voltage is induced in the coil and a current flows in the circuit. (Figure 1)
Part A
If the speed of the magnet is doubled, the induced voltage is ________.
a.) twice as great
b.) four times as great
c.) half as great
d.) unchanged