A mass of 1 slug is attached to a spring whose constant is 5 lb/ft. Initially, the mass is released 1 foot below the equilibrium position with a downward velocity of 7 ft/s, and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to 2 times the instantaneous velocity. (a) Find the equation of motion if the mass is driven by an external force equal to f(t) = 16 cos 2t + 4 sin 2t.

(d²x/dt²) + 2*(dx/dt) + 5*x = 16 cos 2t + 4 sin 2t

Step-by-step explanation:

Given

f(t) = 16 cos 2t + 4 sin 2t

Let m be the mass attached, let k be the spring constant and let β be the positive damping constant. The Newton's Second Law for the system is

m*d²x/dt² = - k*x - β*dx/dt + f(t)

where x(t) is the displacement from the equilibrium position and f(t) is the external force. The equation can be transformed into

(d²x/dt²) + (β/m)*(dx/dt) + (k/m)*x = (1/m)*f(t)      (i)

a) Let's determine the equation of motion. Put  m = 1 slug, k = 5 lb/ft, β = 2 and f(t) = 16 cos 2t + 4 sin 2t into equation (i) to get the differential equation

for x(t):

(d²x/dt²) + (2/1)*(dx/dt) + (5/1)*x = (1/1)*(16 cos 2t + 4 sin 2t)

⇒ (d²x/dt²) + 2*(dx/dt) + 5*x = 16 cos 2t + 4 sin 2t

answer: first option i.e. 'a' is correct.

step-by-step explanation:

since we have given that

number of minutes to wrap a small gift box = 4 minutes

we need to find the number of gifts to be wrapped in 5 minutes

since there is a direct variation, as the number of minutes will increase with   number of gifts to be wrapped.

so, let the number of gifts to be wrapped in 5 minutes be x.

so, it becomes,

hence, first option 'a'   is correct.