In this explainer, we will learn how to solve problems on the motion of two bodies connected by a string passing over a smooth pulley with one of them on an inclined plane.

Consider two bodies connected by a light inextensible string, where the body of mass is supported by a smooth horizontal surface and the body of mass is suspended from the string. The string runs over a smooth pulley that requires negligible force to turn. The following figure shows the forces acting on the bodies, where force is the tension in the string and is the normal reaction force.

The acceleration of each body is determined by its mass and the net force acting on it, according to Newtonβs second law of motion. The tension in the string is constant.

Each body has the same acceleration, given by where is the acceleration due to gravity.

If the surface that supports the body of mass is an inclined plane, then the forces acting on the bodies are those shown in the following figure.

The resultant of the weight of the body of mass , , and the normal reaction force on it, , is , where is the angle above the horizontal of the incline of the plane. The net force on the body parallel to the plane is given by

Let us look at an example in which the acceleration of such a system is determined.

### Example 1: Finding the Acceleration of a System Involving a Smooth Inclined Plane and a Pulley

A body of mass 5 kg rests on a smooth plane inclined at an angle of to the horizontal. It is connected by a light inextensible string passing over a smooth pulley fixed at the top of the plane, to another body of mass 19 kg hanging freely vertically below the pulley. Given that the acceleration due to gravity , determine the acceleration of the system.

### Answer

The following figure shows the forces acting on the bodies and what the forces on the 5 kg mass body due to its weight and the normal reaction on it sum to.

The accelerations of both bodies are equal. The acceleration of the 19 kg mass body is given by

Multiplying the expression by 19, we obtain

The net force on the supported body can be expressed as

Force can also be expressed as

Hence, we have

We now have two equations, (1) and (2), that can be added to give

This simplifies to

To two decimal places, this is 6.59 m/s^{2}.

Let us now look at an example in which the tension in the string is determined.

### Example 2: Finding the Force Acting on the Pulley in a System with an Inclined Plane

Two bodies of equal masses of 7.4 kg are connected by a light inelastic string. One of the bodies rests on a smooth plane inclined at to the horizontal. The string passes over a smooth pulley fixed at the top of the plane, and the other body is left to hang freely vertically below the pulley. Find the force acting on the pulley when the system is released from rest. Take the acceleration due to gravity to be .

### Answer

The force acting on the pulley is the resultant of the tension forces in the strings. The net force on a pulley due to two equal vertical tension forces, and , is shown in the following figure.

In this case, however, one tension force acts vertically downward and the other acts parallel to the inclined plane. The net force due to these two tension forces is, therefore, equivalent to force shown in the following figure.

The angle of inclination of the plane is , hence, the forces on the pulley due to the tensions in the string act as shown in the following figure.

The tension is constant throughout the string, so and have the same magnitude. For the purposes of determining and , this system of forces is equivalent to the system of forces shown in the following figure.

We can define a tension of magnitude , where

The magnitude of the resultant of and is, therefore, given by

The tension in the string can be determined by equating the accelerations of the bodies and, hence, the magnitudes of the forces producing these accelerations, as the masses of the bodies are equal. The forces acting on the bodies are shown in the following figure.

We see from the figure that

This can be rearranged to give

It has been shown that the force on the pulley is given by so, by substituting in our expression for , we have

To two decimal places, this is 130.71 N.

Let us now look at an example in which kinematic equations are used.

### Example 3: Solving an Inclined Plane Pulley System Using Newtonβs Second Law and the Equations of Motion

A body of mass 2.4 kg rests on a smooth plane inclined at an angle of to the horizontal. It is connected by a light inextensible string passing over a smooth pulley, fixed at the top of the plane, to another body of mass 1.6 kg hanging freely vertically below the pulley. When the system was released from rest, the two bodies were on the same horizontal level. Then, 10 seconds later, the string broke. Determine the time taken for the first body to start moving in the opposite direction after the string broke. Take .

### Answer

The initial state of the system is shown in the following figure.

The figure shows that the resultant of the weight of the body of mass and the normal reaction force on the body acts downward parallel to the plane. The force parallel to the plane is the sum of the weight of the body and the normal reaction on the body.

The net force on the supported body is given by

The net force on the suspended body is given by

The acceleration of each body equals the net force on the body divided by its mass. The accelerations of the bodies are equal; hence, the acceleration of the system is given by

We see that and

Adding these expressions, we obtain

Substituting known values, we have

The magnitude of the tension in the string is given by

The magnitude of the sum of the weight of the body and the normal reaction force on it is given by

So, we see that the supported body accelerates upward parallel to the surface and the suspended body descends vertically. We should expect this as the question states that when the tension ceases to act on the body on the surface, it eventually reverses direction. If the body was initially moving downward parallel to the surface, removing the tension force acting on it would not make the body start to move upward parallel to the surface.

In a time of 10 seconds, the velocity of the body is upward along the slope with a magnitude given by

When the tension in the string ceases to act on the body, the body accelerates downward parallel to the surface. With upward parallel to the surface taken as positive, the acceleration is given by

The body starts to move downward parallel to the surface at the instant that it has an instantaneous velocity of zero upward parallel to the surface. The time taken for this velocity change can be determined using the formula

Rearranging to make the subject, we have

Substituting known values, we obtain

Let us now look at an example in which the friction coefficient of a rough inclined surface needs to be found.

### Example 4: Finding the Coefficient of Friction in a System with a Pulley and a Rough Inclined Plane

A body of mass 240 g rests on a rough plane inclined to the horizontal at an angle whose sine is . It is connected, by a light inextensible string passing over a smooth pulley fixed to the top of the plane, to another body of mass 300 g. If the system was released from rest and body descended 196 cm in 3 seconds, find the coefficient of friction between the body and the plane. Take .

### Answer

The resultant of the weight of body and the normal reaction on it acts downward parallel to the plane. This resultant force, , is given by where is the mass of and is the angle that the plane is inclined at. Note that to make the units of mass consistent with the base SI units used for distance and time, the masses of the bodies in grams are converted to masses in kilograms. Thus, substituting , , and , we have

Let us draw a diagram of the system. Bodies and will be connected by a wire, and they will each have a force acting on them due to the tension of the string, with gravity acting in the opposite direction. Additionally, body will have a frictional force acting counter to the direction of motion (i.e., down the slope).

Since the plane is inclined at an angle whose sine is , we can see that the lengths of the sides of the object are in the ratio .

In this question, we need to calculate the coefficient of friction , which is directly related to the magnitude of the frictional force by the formula where is the magnitude of the normal force to the slope. This is given by

Unfortunately, we do not know or , but we do know that the sine of the angle is . As shown in the diagram, this means we know the ratio between the lengths of the sides, and we can use the Pythagorean theorem to find the horizontal distance moved by (which we will denote ):

Hence, we have

Substituting this back into the formula for , we get

To solve this and find , we will need to find , which we can do by considering the other forces in the system.

We note that since the objects are connected, they will accelerate at the
same rate, which can be determined from the motion of
. We are given that body accelerates
from rest and in a time of 3 seconds has a displacement of
196 cm. As the value
of in the question is given as
9.8 m/s^{2}, we
must consider the displacement in
metres rather than
centimetres. So,
the displacement is 1.96 m. The formula for the displacement is
where is the initial velocity, is the
time, and is the acceleration. This can be rearranged to
make the subject, taking to be zero:

Having found the acceleration, this means we can calculate the resultant forces, and , using Newtonβs second law. That is, for ,

and for ,

Each of these resultant forces can also be written as the sum of the forces acting on the bodies. That is,

If we add these two equations together directly, we can cancel out the instances of to get or, rearranged in terms of ,

Then, we can substitute in (3), (4), and (5) to get

Finally, we can get by dividing by 1.8116:

Now, let us look at another example involving a rough inclined plane.

### Example 5: Solving a Rough Inclined Plane Pulley System Using Newtonβs Second Law and the Equations of Motion

A body of mass 162 g rests on a rough plane inclined to the horizontal at an angle whose tangent is . It is connected, by a light inextensible string passing over a smooth pulley fixed to the top of the plane, to another body of mass 181 g hanging freely vertically below the pulley. The coefficient of friction between the first body and the plane is . Determine the distance covered by the system in the first 7 seconds of its movement, given that the bodies were released from rest. Take .

### Answer

We are told that

Hence, when the body supported by the plane moves along the plane, it ascends or descends 4 metres vertically for each 3 metres it travels horizontally. From the Pythagorean theorem, where is the distance traveled along the surface by the body.

Hence, and

The following figure shows all but one of the forces acting on the supported body, where the weight of the body and the normal reaction force on it are shown by their resultant, which acts downward parallel to the plane. is the tension in the string.

The force not shown is the frictional force, , on the supported body. The force is not shown as a frictional force acts in the opposite direction to the net force on a body, and the direction of the net force on the supported body has not been established. Let us now determine it.

The vertically downward force on the body suspended by the string is given by

The force acting parallel to the plane on the body supported by the plane is given by

Dividing by , we obtain

As we see that

Hence, the suspended mass descends and the supported mass moves upward parallel to the plane.

The frictional force on the supported body, is given by the product of the normal reaction force on the body and the coefficient of friction between the body and the surface that supports it. All the forces acting on the supported body act parallel to the plane; hence, we can express the frictional force on the body, , as

The net force on the supported body is, therefore, given by

where is the acceleration of the system.

The net force on the suspended body is given by

Adding the net forces on the bodies given by equations (6) and (7), we obtain

The displacement of the supported body can be determined using the formula

Substituting known values, we find that after 7 seconds of acceleration,

This can be expressed as an integer value by converting it to a value in centimetres, which gives 196 cm.

Let us summarize what we have learned in these examples.

### Key Points

The following points apply to a system of two bodies connected by a light inextensible string, where the body of mass is supported by a smooth plane inclined at an angle above the horizontal and the body of mass is suspended from the string. The string runs over a smooth pulley that requires negligible force to turn, as shown in the following figure.

- The net force on the suspended body equals the sum of the tension in the string and the weight of the body. It is given by where is the tension in the string and is the acceleration due to gravity.
- The net force on the supported body equals the sum of the tension in the string and the component of the weight of the body acting parallel to the plane. This is given by where is the tension in the string and is the acceleration due to gravity.
- The accelerations of the suspended and the supported bodies are both equal to the net forces on them divided by their masses, and both accelerations are equal. Then, we have where is the tension in the string and is the acceleration due to gravity.
- If the inclined plane that supports the body of mass is rough, the acceleration of the system is given by where is the coefficient of friction of the body with the plane. The frictional force on the body of mass may be either in the same direction as or in the opposite direction to the tension in the string, depending on whether the body moves upward or downward parallel to the plane.