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Chapter 8: Problem 45

Find the \(x\) - and \(y\) -coordinates of the center of mass of the flat triangular plate of height \(H=17.3 \mathrm{~cm}\) and base \(B=10.0 \mathrm{~cm}\) shown in the figure.

Short Answer

Expert verified
Answer: The coordinates of the center of mass of the flat triangular plate are (5, 5.77) cm.

Step by step solution

01

Identify the coordinates of the vertices of the triangle

We can consider the base of the triangle to be along the x-axis and one vertex of the triangle at the origin. Then, the vertices of the triangle are A(0, 0), B(10, 0), and C(5, 17.3).
02

Calculate the x-coordinate of the center of mass

The x-coordinate of the center of mass of a flat triangular plate is given by the average of the x-coordinates of its vertices. So, we can find the x-coordinate of the center of mass using the formula: \(x_{c} = \frac{A_x + B_x + C_x}{3}\) Now substitute the x-coordinates of the vertices A(0), B(10), and C(5) in the above formula: \(x_{c} = \frac{0 + 10 + 5}{3} = \frac{15}{3} = 5 \mathrm{~cm}\)
03

Calculate the y-coordinate of the center of mass

Similarly, the y-coordinate of the center of mass of a flat triangular plate is given by the average of the y-coordinates of its vertices. So, we can find the y-coordinate of the center of mass using the formula: \(y_{c} = \frac{A_y + B_y + C_y}{3}\) Now substitute the y-coordinates of the vertices A(0), B(0), and C(17.3) in the above formula: \(y_{c} = \frac{0 + 0 + 17.3}{3} = \frac{17.3}{3} = 5.77 \mathrm{~cm}\)
04

State the coordinates of the center of mass

The coordinates of the center of mass of the flat triangular plate are \((x_c, y_c) = (5, 5.77)\).

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Most popular questions from this chapter

A uniform log of length \(2.50 \mathrm{~m}\) has a mass of \(91.0 \mathrm{~kg}\) and is floating in water. Standing on this log is a \(72.0-\mathrm{kg}\) man, located \(22.0 \mathrm{~cm}\) from one end. On the other end is his daughter \((m=20.0 \mathrm{~kg})\), standing \(1.00 \mathrm{~m}\) from the end. a) Find the center of mass of this system. b) If the father jumps off the log backward away from his daughter \((v=3.14 \mathrm{~m} / \mathrm{s}),\) what is the initial speed of \(\log\) and child?

An artillery shell is moving on a parabolic trajectory when it explodes in midair. The shell shatters into a very large number of fragments. Which of the following statements is true (select all that apply)? a) The force of the explosion will increase the momentum of the system of fragments, and so the momentum of the shell is not conserved during the explosion. b) The force of the explosion is an internal force and thus cannot alter the total momentum of the system. c) The center of mass of the system of fragments will continue to move on the initial parabolic trajectory until the last fragment touches the ground. d) The center of mass of the system of fragments will continue to move on the initial parabolic trajectory until the first fragment touches the ground. e) The center of mass of the system of fragments will have a trajectory that depends on the number of fragments and their velocities right after the explosion.

A system consists of two particles. Particle 1 with mass \(2.0 \mathrm{~kg}\) is located at \((2.0 \mathrm{~m}, 6.0 \mathrm{~m})\) and has a velocity of \((4.0 \mathrm{~m} / \mathrm{s}, 2.0 \mathrm{~m} / \mathrm{s}) .\) Particle 2 with mass \(3.0 \mathrm{~kg}\) is located at \((4.0 \mathrm{~m}, 1.0 \mathrm{~m})\) and has a velocity of \((0,4.0 \mathrm{~m} / \mathrm{s})\) a) Determine the position and the velocity of the center of mass of the system. b) Sketch the position and velocity vectors for the individual particles and for the center of mass.

A man with a mass of 55 kg stands up in a \(65-\mathrm{kg}\) canoe of length \(4.0 \mathrm{~m}\) floating on water. He walks from a point \(0.75 \mathrm{~m}\) from the back of the canoe to a point 0.75 m from the front of the canoe. Assume negligible friction between the canoe and the water. How far does the canoe move?

A man standing on frictionless ice throws a boomerang, which returns to him. Choose the correct statement::: a) Since the momentum of the man-boomerang system is conserved, the man will come to rest holding the boomerang at the same location from which he threw it. b) It is impossible for the man to throw a boomerang in this situation. c) It is possible for the man to throw a boomerang, but because he is standing on frictionless ice when he throws it, the boomerang cannot return. d) The total momentum of the man-boomerang system is not conserved, so the man will be sliding backward holding the boomerang after he catches it.
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