Answer:
the required value is 
Explanation:
Given that,
mass, m = 1kg
spring constant k = 21N/M
damping force = 

By Newtons second law ,
The diffrential equation of motion with damping is given by

substitute the value of m =1kg, k = 21N/M, and 


suppose the equation of the form
,
and the auxilliary equation is given by

The general solution for the above differential equation is

Derivate with respect to t

(a)
since time is 0 then mass is one meter below
so x(0) = 1
Also it start from rest , that implies , velocity is 0 and time is 0

substitute the initial condition


Solve the above equation to get C₁ and C₂

substitute for C₁ and C₂ in general solution

Thus the required value is 
When smoke particles pass between the source of radiation and the detector the drop in radiation will be sufficient to tigger the alarm. <span />
The question is incomplete! The complete question along with answer and explanation is provided below.
Question:
A 0.5 kg mass moves 40 centimeters up the incline shown in the figure below. The vertical height of the incline is 7 centimeters.
What is the change in the potential energy (in Joules) of the mass as it goes up the incline?
If a force of 1.0 N pulled up and parallel to the surface of the incline is required to raise the mass back to the top of the incline, how much work is done by that force?
Given Information:
Mass = m = 0.5 kg
Horizontal distance = d = 40 cm = 0.4 m
Vertical distance = h = 7 cm = 0.07 m
Normal force = Fn = 1 N
Required Information:
Potential energy = PE = ?
Work done = W = ?
Answer:
Potential energy = 0.343 Joules
Work done = 0.39 N.m
Explanation:
The potential energy is given by
PE = mgh
where m is the mass of the object, h is the vertical distance and g is the gravitational acceleration.
PE = 0.5*9.8*0.07
PE = 0.343 Joules
As you can see in the attached image
sinθ = opposite/hypotenuse
sinθ = 0.07/0.4
θ = sin⁻¹(0.07/0.4)
θ = 10.078°
The horizontal component of the normal force is given by
Fx = Fncos(θ)
Fx = 1*cos(10.078)
Fx = 0.984 N
Work done is given by
W = Fxd
where d is the horizontal distance
W = 0.984*0.4
W = 0.39 N.m
-40 c = -40 f but k would be 233.15