Question

A 9.20-kg hanging object is connected by a light, inextensible cord over a light, frictionless pulley to a 5.00-kg block that is sliding on a flat table. Taking the coefficient of kinetic friction as 0.158, find the tension in the string.

Answer 1

Concepts and reason

The concepts used to solve this problem are friction force, tension, force due to gravity, and free body diagram.

First, draw the free body diagram of each block. After that apply the force equilibrium

equation on the block and . Next equate the both force equilibrium equation to

calculate the acceleration of the block. And finally calculate the tension in the string by

substitute the calculated acceleration in any of the force equilibrium equation.

The mass of block is greater than mass of block. So, the block slides on table

due to the gravity that pulls block of mass in downward direction since both blocks are attached with an inextensible string. And due to motion of block that have mass on table a non-conservative kinetic friction force acts in opposite

direction of motion.

Fundamentals

Kinetic friction force is a force that acts between moving surfaces. The force of kinetic friction is times the normal force on an object, and is expressed in units of Newton.

Here is the coefficient of kinetic friction, and is normal force that acts in upward direction.

The force due to gravity between a mass and the Earth is the object's weight. Mass is considered a measure of an object's inertia, and its weight is the force exerted on the object in a gravitational field. On the surface of the Earth, the two forces are related by the acceleration due to gravity is,

Here, is the mass of body, and is gravitational acceleration.

Tension arises in the use of ropes, cables or string to transmit a force. Rather a force is exerted on the string, which transmits that force to the block. The force experienced by the block from the string is called the tension force.

Free-body diagram is a diagram which shows the relative magnitude and direction of all forces acting upon an object in a given situation.

For block 1:

The below figure shows the free body diagram of block ${m_1}$

Refer the above free-body diagram of the block ${m_1}$ .

The net force on block in horizontal direction

$T - {f_k} = {m_1}a$

$T = {m_1}a + {f_k}$

Substitute ${\mu _k}{m_1}g$ for ${f_k}$ .

$T = {m_1}a + {\mu _k}{m_1}g$ …… (1)

For block 2:

The below figure shows the free body diagram of block ${m_2}$

Refer the free-body diagram of the block ${m_2}$ .

The net force on block in vertical direction is,

${m_2}g - T = {m_2}a$

$T = {m_2}g - {m_2}a$ …… (2)

Tension in the string for both blocks is same so, from equation (1) and (2).

$\begin{array}{l}\\{m_2}g - {m_2}a = {m_1}a + {\mu _k}{m_1}g\\\\{m_1}a + {m_2}a = {m_2}g - {\mu _k}{m_1}g\\\\a = \frac{{{m_2}g - {\mu _k}{m_1}g}}{{{m_1} + {m_2}}}\\\end{array}$

Substitute $9.20{\rm{ kg}}$ for ${m_2}$ , $9.8{\rm{ m/}}{{\rm{s}}^{\rm{2}}}$ for $g$ , $0.158$ for ${\mu _k}$ , $5.00{\rm{ kg}}$ for ${m_1}$ .

$\begin{array}{c}\\a = \frac{{\left( {9.20{\rm{ kg}}} \right)\left( {9.8{\rm{ m/}}{{\rm{s}}^{\rm{2}}}} \right) - \left( {0.158} \right)\left( {5.00{\rm{ kg}}} \right)\left( {9.8{\rm{ m/}}{{\rm{s}}^{\rm{2}}}} \right)}}{{\left( {5.00{\rm{ kg}}} \right) + \left( {9.20{\rm{ kg}}} \right)}}\\\\ = \frac{{\left( {90.16 - 7.742} \right){\rm{kg}} \cdot {\rm{m/}}{{\rm{s}}^2}}}{{\left( {14.20{\rm{ kg}}} \right)}}\\\\ = \frac{{\left( {82.418} \right){\rm{kg}} \cdot {\rm{m/}}{{\rm{s}}^2}}}{{\left( {14.20{\rm{ kg}}} \right)}}\\\\ = 5.80{\rm{ m/}}{{\rm{s}}^2}\\\end{array}$

Now calculate the tension of the string.

Substitute $9.20{\rm{ kg}}$ for ${m_2}$ , $9.8{\rm{ m/}}{{\rm{s}}^{\rm{2}}}$ for $g$ ,and $5.80{\rm{ m/}}{{\rm{s}}^2}$ forin equation (2).

$\begin{array}{c}\\T = \left( {9.20{\rm{ Kg}}} \right)\left( {9.8{\rm{ m/}}{{\rm{s}}^{\rm{2}}}} \right) - \left( {9.20{\rm{ Kg}}} \right)\left( {5.80{\rm{ m/}}{{\rm{s}}^{\rm{2}}}} \right)\\\\ = 90.16{\rm{ N}} - 53.86{\rm{ N}}\\\\ = 36.8{\rm{ N}}\\\end{array}$

Ans:

The tension in string is $36.8{\rm{ N}}$ .