Question

1. A public art installation consists of three 25-kg glass sculptures hung side-by-side with steel wires of length 1.00 m, 2.00 m and 3.00 m. If the wires all have the same diameter, which one stretches the greatest distance?A All three wires will stretch the same distance.B The 1.00-m wire will stretch the greatest distance.C The 2.00-m wire will stretch the greatest distance.D The 3.00-m wire will stretch the greatest distance.E There is not enough information to say.

2. A solid uniform cylinder is rolling without slipping. What fraction of this cylinder's kinetic energy is rotational? (Note: the moment of inertia of a cylinder of mass M and radius R rotating about its central axis is 0.5MR².)  A 1/3B 2/3C 1/2D 1/4E 3/4

Answer 1

1. For steel wire, the Young's Modulus (Y) is given by,

where,

Given,

• All the three sculptures are of 25kg. So the weight of all of them is the same.
• All of them are of the same diameter. So all of them will have the same radius and hence the area cross-section.
• Since all the three wires are of steel, the Youngs modulus will also be the same.

The only thing which is different is the length. hence from the equation of Young's Modulus, the extension is given by,

Since F, A and Y are constants in this case,

ie, extension produced is directly proportional to the original length of the wire.

Hence the wire 3m wire will produce the greatest extension. So the correct option is option D

2. The translational kinetic energy of the solid uniform cylinder is given by

where,

The rotational kinetic energy of the solid uniform cylinder is given by

where,

Hence the total kinetic energy of the solid uniform cylinder is given by

For an object that is rolling without slipping,

The fraction of total kinetic energy that is rotational is

For the solid uniform cylinder, the moment of inertia is given by,

Therefore, the fraction of total kinetic energy that is rotational is given by

So one third fraction of total kinetic energy is rotational. Hence option a is correct