A brick of mass 4 kg hangs from the end of a spring. When the brick is at rest, the spring is stretched by 7 cm. The spring is then stretched an additional 2 cm and released. Assume there is no air resistance. Note that the acceleration due to gravity, g, is g

a) d^2x/dt^2 = -1.4 * x

b) x = -1.4 * cos1.2*t

Explanation:

a)

In the equilibrium state, the force of gravity like stretching equals each other:

m*g = -k*x

Being the mass of the brick of 4 kg, which causes a 7 cm stretch being in balance:

4*9.8 = -k * (-7)

39.2 = 7*k

k = 5.6

movement of the mass that is attached to the spring is equal to:

d^2x / dt^2 = -w^2 * x, where w^2 = k/m = 1.4

the differential equation is as follows:

d^2x/dt^2 = -1.4 * x

When t = 0, the rope is stretched 1.4 cm, therefore, the initial condition is equal to:

x(0) = -1.4 and x´(0) = 0

b) to solve the differential equation is equal to:

x = C1 * cos(w*t) + C2 * sin(w*t)

replacing the value of w, w = (1.4)^1/2 = 1.2

when t = 0

-1.4 = C1 * cos0 + C2 * sin0

C1 = -1.4

x´(0) = -1.2 * C1 * sin0 + 1.2 * C2 * cos0

0 = 2.2 * C2

C2 = 0

The solution to the equation is equal to:

x = -1.4 * cos1.2*t

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answer: the correct answer is 1.

explanation: there are various process in which a radioactive nuclei decays:

1) alpha decay: in this process, a heavier nuclei decays into lighter nuclei by releasing alpha particle. the mass number is reduced by 4 units.

2) electron capture: in this decay process, a parent nuclei absorbs an electron and gets converted into a neutron. simply, a proton and an electron combines together to form a neutron. mass number does not change in this process.

3) gamma ray emission: it is a decay process in which an unstable nuclei gives excess energy by a spontaneous electromagnetic process. this decay releases . this process does not change the mass number.

4) positron emission: it is a type of decay process, in which a proton gets converted to neutron and an electron neutrino. this is also known as -decay. in this the mass number remains same.

from the above information, we see that only alpha-decay is producing a change in the mass number of the nucleus.