need help solving these IT 1. Two spheres are cut from a certain uniform rock. O

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need help solving these

IT 1. Two spheres are cut from a certain uniform rock. One has 9 cm. The mass of the other radius is half times smaller. Find its 2. A particle moves along the x axis according to the equation x 2 4t +1, where x is in meters and t is in seconds, Att 4 s, find (a) the position of the particle, (b) its velocity, and (c) its acceleration. 3. A traffic light weighing 122 N hangs from a cable tied to two other cables fastened to a support, as in Figure. The upper cables make angles 37.0" and 53.0" with the horizontal. These upper the are not as strong as the vertical cable, of tension in exceeds 100 Does and will break if the them traffic light remain hanging in this situation, or will one of the cables break?

Explanation / Answer

(1) Uniform rock means equal density.

Mass of other = (1/2)*Mass of orginal

=> V2 = (/2)V1

Now, V = (4/3)*pi*R^3

=> Radius is proportional to V^1/3

So, R2 = (R1)^1/3 = 9^1/3 = 2.08 cm.

So, radius of other sphere = 2.08 cm

(2) (a) put t = 4 in the given expression -

x(4) = 4^2 - 4*4 + 1 = 1 m

(b) v = dx/dt = 2t - 4

v(4) = 2*4 - 4 = 4 m/s

(c) a = dv/dt = 2

so, a(4) = 2 m/s^2

(3) Consider T1 and T2 are the tensions in the two cables.

Now, resolving the forces -

T1*cos37 = T2*cos53

=> T1*0.80 = T2*0.60

=> T2 = 1.3*T1-------------------------------------------------(i)

Further, T1*sin37 + T2*sin53 = 122

=> T1*0.6 + 1.3*T1*0.8 = 122

=> T1 = 74 N

and T2 = 1.3*74 = 96.2 N

So, tension in both the cables will not exceed 100 N.

So, the cable will not break.

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