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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

Two point masses are located in the same plane. The distance from mass 1 to the center of mass is \(3.0 \mathrm{~m} .\) The distance from mass 2 to the center of mass is \(1.0 \mathrm{~m} .\) What is \(m_{1} / m_{2},\) the ratio of mass 1 to mass \(2 ?\) a) \(3 / 4\) c) \(4 / 7\) e) \(1 / 3\) b) \(4 / 3\) d) \(7 / 4\) f) \(3 / 1\)

A uniform chain with a mass of \(1.32 \mathrm{~kg}\) per meter of length is coiled on a table. One end is pulled upward at a constant rate of \(0.47 \mathrm{~m} / \mathrm{s}\). a) Calculate the net force acting on the chain. b) At the instant when \(0.15 \mathrm{~m}\) of the chain has been lifted off the table, how much force must be applied to the end being raised?

A cart running on frictionless air tracks is propelled by a stream of water expelled by a gas-powered pressure washer stationed on the cart. There is a \(1.00-\mathrm{m}^{3}\) water tank on the cart to provide the water for the pressure washer. The mass of the cart, including the operator riding it, the pressure washer with its fuel, and the empty water tank, is \(400 . \mathrm{kg} .\) The water can be directed, by switching a valve, either backward or forward. In both directions, the pressure washer ejects \(200 .\) L of water per min with a muzzle velocity of \(25.0 \mathrm{~m} / \mathrm{s}\). a) If the cart starts from rest, after what time should the valve be switched from backward (forward thrust) to forward (backward thrust) for the cart to end up at rest? b) What is the mass of the cart at that time, and what is its velocity? (Hint: It is safe to neglect the decrease in mass due to the gas consumption of the gas-powered pressure washer!) c) What is the thrust of this "rocket"? d) What is the acceleration of the cart immediately before the valve is switched?

A projectile is launched into the air. Part way through its flight, it explodes. How does the explosion affect the motion of the center of mass of the projectile?

You are piloting a spacecraft whose total mass is \(1000 \mathrm{~kg}\) and attempting to dock with a space station in deep space. Assume for simplicity that the station is stationary, that your spacecraft is moving at \(1.0 \mathrm{~m} / \mathrm{s}\) toward the station, and that both are perfectly aligned for docking. Your spacecraft has a small retro-rocket at its front end to slow its approach, which can burn fuel at a rate of \(1.0 \mathrm{~kg} / \mathrm{s}\) and with an exhaust velocity of \(100 \mathrm{~m} / \mathrm{s}\) relative to the rocket. Assume that your spacecraft has only \(20 \mathrm{~kg}\) of fuel left and sufficient distance for docking. a) What is the initial thrust exerted on your spacecraft by the retro-rocket? What is the thrust's direction? b) For safety in docking, NASA allows a maximum docking speed of \(0.02 \mathrm{~m} / \mathrm{s}\). Assuming you fire the retro-rocket from time \(t=0\) in one sustained burst, how much fuel (in kilograms) has to be burned to slow your spacecraft to this speed relative to the space station? c) How long should you sustain the firing of the retrorocket? d) If the space station's mass is \(500,000 \mathrm{~kg}\) (close to the value for the ISS), what is the final velocity of the station after the docking of your spacecraft, which arrives with a speed of \(0.02 \mathrm{~m} / \mathrm{s}\) ?
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