A solid cube of known mass (mc-1.53 kg) is initially at rest on a steep slope, a

Question

A solid cube of known mass (mc-1.53 kg) is initially at rest on a steep slope, as shown here. The cube is tethered to a system of ideal pulleys (and the wire portion attached directly to the cube is parallel to the slope) Friction (,-0.876; ,-0.419) is available between the cube and the slope. A large, empty bucket is hanging at rest as shown. Static friction is just barely sufficient to keep the cube from slipping along the slope Otherdata: =20.00 a. Water is gradually poured into the bucket until the slope is exerting zero 9.80 m/s, friction force; the cube remains at rest. Calculate the mass of the water b. bucket More water is gradually poured into the bucket until the cube is again ready to slip (but still, it doesn't). Calculate the weight of the cube Now one more drop of water (say, 0.00002 kg) is added to the bucket, and the cube begins to slide. Calculate the weight of the cube during the first moments of its slide (ie, before has changed measureably). cube c.

Explanation / Answer

given, cube mass mc = 1.53 kg

coefficient of static friction Ks = 0.876

coefficient of kinetic firction K = 0.419

a. theta = 20 deg

g = 9.8 m/s/s

let mass of water be m

tension in strings be T

then from force balance

mg = 2Tcos(theta)

T = mcgcos(theta)

mg = 2*mc*gcos(theta)*cos(theta)

m = 2*mc*cos(theta)cos(theta)

m = 2.702 kg

b. when the cube is ready to slip

the firction on the cube is downwards along the inlcine and is static friction

let mass of water be m

from force balance

mg = 2Tcos(theta)

T = mc*g*cos(theta) + Ks*mc*g*sin(theta)

hence

mg = 2*mc*g*(cos(theta) + Ks*sin(theta))cos(theta)

m = 3.5635 kg

weight of water = 34.95 N

c. now, m = 3.5635 + 0.00002 = 3.56352 kg

hence from force balance

mg - 2Tcos(theta) = m*a

T - mc*g*cos(theta) - K*mc*g*sin(theta) = mc*a

hence

mg - 2*(mc*a + mc*g*cos(theta) + K*mc*g*sin(theta))cos(teta) = ma

34.95875 - 1.879(1.53a + 16.255 ) = 3.56352*a

a = 0.6858 m/s/s

weight of cube = mc*a = 1.049285 N

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