Question

(a) A constant force < 31, -13, 36 > N acts through a displacement < 0.17, 0.34, -0.24 > m. How much work does this force do?
Work =           ?J

(b) An object with mass 7 kg moves from a location < 28, 28, -43 > m near the Earth's surface to location < -34, 16, 45 > m. What is the change in the potential energy of the system consisting of the object plus the Earth?
Change of potential energy =     ?J

(c) A spring whose stiffness is 900 N/m has a relaxed length of 0.59. If the length of the spring changes from 0.48 m to 0.86 m, what is the change in the potential energy of the spring?
Change of potential energy =      ?J

(d) You observe someone pulling a block of mass 33 kg across a low-friction surface. While they pull a distance of 6 m in the direction of motion, the speed of the block changes from 3 m/s to 5 m/s. Calculate the magnitude of the force exerted by the person on the block.
F =     ?N

(e) What was the change in internal energy (chemical energy plus thermal energy) of the person pulling the block?
Change in internal energy =     ?J

Answer 1

Solution:

a) Work done = W = F.d

F = 31 i - 13 j + 36 k and d = 0.17 i + 0.34 j - 0.24 k

F.d = ( 31 i - 13 j + 36 k ).(0.17 i + 0.34 j - 0.24 k) = (31*0.17 )+(-13*0.34)+(36 *-0.24)

= -7.79 joules

b) Initial position =h1 = 28 i +28i -43k m; final height = h2 = -34i+16j+45k m

Displacement between the two points = h2-h1 =(-34 -28)i +(16-28)j +(45-(-43)k m

=> h= -62 i -12 j +88 k = (-62)^2 + (-12)^2+(88)^2 = 108 m

Gravitational potential energy = mgh = (7)(9.8)(108)

= 7430 joules

c) spring stiffness = k = 900 N/m

relaxed length = L=  0.59. m

When its length becomes 0.48m, it is compression .

Potential energy due to compression = 1/2 k( x2 - x1) ^2= 1/2 (900)(0.48 -0.59) ^2

= 5.4445 joules

When it stretches to 0.86m, Potential energy = 1/2 (900)(0.86 - 0.59)^2

= 32.805 j

Change in potential energy = 32.805 - 5.4445

= 27.4 joules

d) Work done = Change in Kinetic Energy

F.d = 1/2 m(vf^2 - vi^2)

F = m (vf^2 - vi^2) /2d = 33 *(5^2 - 3^2) / 2*6

= 44 N

e) Change in internal energy of the person pulling the block = F.d = change in kinetic energy

= -44 * 6 = -264 J