FORCE AND ACCELERATION: Mechanical Advantage and Tension Within Pulley Systems

The law of conservation of energy states that the total energy of an isolated system remains constant. The SI unit of energy is the joule ( J ), and it’s base-unit composition is kg*m²/s². Energy is the currency needed to perform work, and work is performed upon an object when an applied force moves it through some distance:

W = Fd

Force ( F ) = ma, where m = mass in kilograms ( kg ), and a = acceleration. The base-units of acceleration are meters per second squared ( m/s² ). Since energy cannot be created or destroyed, the work put into a system of pulleys must equal the work output of the system:

F₁d₁ = F₂d₂

Please note that the input forces in the diagrams that follow represent the minimum force values needed to maintain the suspended object’s stationary position. Work is only done when the applied force pulls the suspended object upward. If the force needed to lift a suspended object is equal to the weight of the object ( F_w ), the distance the belt is pulled downward will equal the distance the object rises upward. Within the diagrams that follow, friction is considered to be negligible. Additionally, the pulleys are assumed to be massless, and the moment of inertia ( I ) created when belts exert a force upon a pulley is considered to be negligible as well:

In the system above, F₁d₁ = F₂d₂, and F = F_w. Since the force input needed to lift the 100 N object is 100 N, no mechanical advantage ( MA ) is gained when the belt is pulled downward. We will soon see, however, that placing multiple points of resistance within the system will decrease the input force needed to lift the suspended object. Generally speaking, the mechanical advantage of a system increases in proportion to the number of pulleys around which the system’s belt is wrapped. Mechanical advantage is mathematically defined by the following ratio:

MA = ( F_out / F_in )

Additionally, since W_in = W_out, the law of conservation of energy may be expressed as follows: 

F_ind_in = F_outd_out

As a consequence, mechanical advantage may also be expressed as follows:

MA = ( F_out / F_in ) = ( d_in / d_out )

When a system’s mechanical advantage lessens the input force needed to lift a suspended object upward, the upward distance travelled by the body will be proportionally less than the length of the belt that is pulled downward.

In the system below, the addition of a pulley to the system gives rise to two lengths of belt that resist the 100 N downward pull on the system. According to Newton’s Third Law of Motion, for every force, there is an equal and opposite force. The force pair within the lengths of belt that emerge from each side of a pulley is referred to as tension ( T ).

Analysis of pulley systems is simplified when the net forces acting upon isolated system components are identified in free-body diagrams. The system below consists of two belts. The first belt wraps around both pulleys before its terminal end is secured to the uppermost pulley. The second pulley connects the lower pulley to the suspended object:

Tension is distributed equally throughout each belt. As a consequence, each isolated segment of a belt represents a force pair that opposes motion in the horizontal and vertical directions. Moving in the clockwise direction with the first belt shows that F = T = T. If the first pulley is isolated, the force vectors in the downward direction have a magnitude of 3T. As a consequence, the belt that secures the uppermost pulley sustains 3T Newtons of force. The lower pulley has 2T Newtons of force opposing the downward pull of the suspended object; therefore, 2T = 100 N, and T = 50 N. Notice that the sum of the tension holding the suspended body in place ( 2T ) plus the input tension of F = T equals 3T. Thus, the downward pull of the system is equal and opposite to the tension in the uppermost belt. The mechanical advantage of the system as follows:

MA = ( F_out / F_in ) = ( 100 N / 50 N ) = 2

Additionally, MA = ( F_out / F_in ) = ( d_in / d_out )

2 = ( d_in / d_out )

2d_out = d_in

d_out = ½ d_in

For every foot downward the input belt is pulled, the suspended object will rise ½ foot. As pulley diagrams become more complicated, similar use of free-body diagrams are used to determine the mechanical advantage provided by the pulleys within the system. The following system has three pulleys within it. Each of these pulleys adds inertia to the system, and we can easily predict that the mechanical advantage value will be equal to 3: 

3T = 100 N, so T = 33.3 N

MA = ( 100 N / 33.3 ) = 3

Additionally, for every meter of the belt pulled downward by the input force, the suspended body will rise ⅓ of the length. The system below has four pulleys, and the mechanical advantage of the system is 4:

4T = 100 N, so T = 25 N

MA = ( 100 N / 25 N ) = 4

For every meter the belt is pulled downward by the input force, the box rises ¼ of a meter.