6. A mass of 0.4000 kg is raised by a vertical distance of 0.4500 m in the earth

Question

6. A mass of 0.4000 kg is raised by a vertical distance of 0.4500 m in the earth's gravitational field. What is the change in its gravitational potential energy? This same mass is lowered by a vertical distance of 0.3500 m in the earth's gravita- tional field. What is the change in its gravitational potential energy? 7. A spring of spring constant k A mass of 0.5000 kg is placed on the end of the spring and supported by hand at a point so that the displacement of the spring is 0.2500 m. The mass is suddenly released and allowed to fall. At the lowest position of the mass, what is the dis- placement of the spring from its equilibrium position? (Hint: In solving this prob- lem you will have to solve a quadratic equation, and one of the solutions will be the original 0.2500-m displacement.) 8. The same spring described in question 7 again has a mass of 0.5000 kg on the end of the spring. It is pulled down by hand and held so that the displacement of the spring from its equilibrium position is 0.7000 m. If the system is suddenly released and the mass is allowed to rise, what is the displacement of the spring when the mass is at its highest position? (Hint: In solving this problem you will have to solve a quadratic equation, and one of the solutions will be the original 0.7000-m displacement.)

Explanation / Answer

1) Any object which is raised against gravity has increase in its potential energy

at bottom, P.E = 0,

at h = 0.45 m, P.E = 0.4*9.8*0.45

P.E = 1.764 joules,

change in P.E = final - initial

change in P.E = 1.764 joules

Now, when it is lowered, there is a decrease in P.E

2) use the following equation

mg (x2 - x1) = 1/2*k(x2^2 - x1^2)

we need to solve for x2

0.5*9.8(x2 - 0.25) = 1/2*8.750*(x2^2 - x1^2)

4.9x2 - 1.225 = 4.375x2^2 - 0.2734

we have a quadratic equation,

4.375x2^2 - 4.9x2 + 0.9516 = 0

solving it, we get

x2 = 0.8699 m , 0.25 m

therefore, displacement x = 0.8699 m

3) same method as above, in this object is rising.

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