Mass and Acceleration

Although sometimes you may not realise it, forces act on you all of the time. The force of gravity pulls you downwards, and the Earth's surface pushes back up on you with an equal and opposite force. On a windy day, you will feel a force in the direction of the wind due to the air particles buffeting against you. When the forces acting on an object are imbalanced, the object's motion changes - it accelerates. The size of this acceleration depends on the object's mass. For example, it is easier to lift a pencil than a whole desk. In this article, we will discuss the relationship between mass and acceleration and explore the tools we can use to describe it.

Mass and acceleration formula

In physics, you will come across the mass and acceleration of objects all of the time. It is very important to understand exactly what the words mean, how to use them, and how mass and acceleration are related.

Mass

The SI unit for mass is \( \mathrm{kg} \). The mass of an object does not only depend on its size (volume) but also on its density. The mass of an object in terms of its density is given by the formula:

$$m=\rho V,$$

where \( \rho \) is the density of the material of the object in \( \mathrm{kg}/\mathrm{m^3} \) and \( V \) is its volume in \( \mathrm{m^3} \). We can see from the formula that, for objects of the same volume, a higher density will lead to a higher mass. The formula can be rearranged to find an expression for density as

$$\rho=\frac mV.$$

Mass and Weight

You must not confuse the mass of an object with its weight, they are very different things! An object's mass is always constant, no matter where it is, whereas an object's weight changes depending on the gravitational field it is in and its position in that gravitational field. Also, mass is a scalar quantity - it only has a magnitude - whereas weight is a vector quantity - it has a magnitude and a direction.

The weight of an object is measured in \( \mathrm N \) and is given by the formula

$$W=mg,$$

where \( m \) is again the object's mass and \( g \) is the gravitational field strength at the point where the object is measured in \( \mathrm m/\mathrm{s^2} \), which are the same units as for acceleration. As you can see from the formula, the larger an object's mass, the larger its weight will be. In most practice problems, you will have to use the gravitational field strength on the Earth's surface, which is equal to \( 9.8\,\mathrm m/\mathrm{s^2} \).

Acceleration

The SI unit for acceleration is \( \mathrm m/\mathrm{s^2} \). The acceleration of an object can be calculated with the formula

$$a=\frac{\Delta v}{\Delta t},$$

where \( \Delta v \) is the change in velocity (measured in \( \mathrm m/\mathrm s \)) in a time interval \( \Delta t \) measured in \( \mathrm s \).

Notice that the formula for acceleration includes velocity, and not speed. As you might already know, the velocity of an object is its speed in a given direction. This means that the direction in which the speed changes is important when calculating acceleration, as acceleration also has direction. Both velocity and acceleration are vector quantities. An object that slows down (decelerates) has a negative acceleration.

Newton's second law

In order to accelerate an object, a force is needed. The resultant force is the force found by adding up all the different forces acting on a body. This needs to be done vectorially - each force arrow is connected from head to tail.

Fig. 2 - Forces must be added together vectorially.

Newton's famous second law states:

This explanation of Newton's law is quite long and can often be confusing, but fortunately, the law is also perfectly summed up by the equation

$$F=ma,$$

where \( F \) is the resultant force on an object in \( \mathrm N \), \( m \) is the object's mass in \( \mathrm{kg} \), and \( a\) is the object's acceleration in \( \mathrm m/\mathrm{s^2} \).

Let's see how this formula is equivalent to the statement above. Newton's second law says that the acceleration of an object is directly proportional to the resultant force. We know that the mass of an object is constant, so the formula shows that the resultant force is equal to the acceleration multiplied by a constant, meaning that the force and the acceleration are directly proportional.

The law also states that the acceleration of an object is in the same direction as the resultant force. We can see how the formula also shows this by remembering that force and acceleration are both vectors, so they both have a direction, whereas mass is a scalar, which can simply be described by its magnitude. The formula states that force is equal to acceleration multiplied by a constant, so there is nothing to change the direction of the acceleration vector meaning that the force vector points in the same direction as the acceleration.

Fig. 3 - A force points in the same direction as the acceleration that it causes.

Finally, Newton's second law says that the acceleration of an object is directly proportional to its mass. The formula can be rearranged to

$$a=\frac Fm,$$

which shows that, for a given force, the acceleration of an object is inversely proportional to its mass. If you increase the mass of the object to which the force is being applied, its acceleration will decrease, and vice versa.

Inertial mass

The rearranged version of Newton's second law leads us to the concept of inertial mass.

The inertial mass of an object is the resistance to acceleration caused by any force whereas the gravitational mass of an object is determined by the force acting on an object in a gravitational field. Despite their different definitions, these two quantities have the same value. You can think of the mass of an object as its resistance to a change in motion. The greater the mass of an object, the more force is required to give it a certain acceleration and hence increase its velocity by a given amount.

Investigating the effect of mass on acceleration

The rearranged version of Newton's second law can be used to investigate the effect of mass on acceleration. We stated Newton's law in equation form in the last section, but how do we know this is true? Do not take our word for it, let's instead test it through an experiment!

Newton's second law can be rearranged to

$$a=\frac Fm.$$

We want to investigate how changing the mass of an object affects the acceleration of that object for a given force - we keep the force constant and see how the other two variables change. There are several ways to do this but we will take just one example.

An experimental setup is shown above. Place a pulley on the end of a bench and keep it in place by using a clamp. Pass a string over the pulley. Tie a mass onto the end of the string hanging off the bench, and then tie a cart onto the opposite end of the string. Set up two light gates for the cart to pass through and a data logger to calculate the acceleration. Before starting the experiment, use some weighing scales to find the mass of the cart.

For the first reading, place the empty cart in front of the first light gate, release the mass hanging from the pulley and let it fall to the floor. Use the data logger to calculate the acceleration of the cart. Repeat this three times and take a mean of the accelerations to get a more accurate result. Then place a mass inside the cart (\(100\,\mathrm{g}\) for example) and repeat the process. Continue to add weights to the cart and measure the acceleration each time.

Evaluation of mass and acceleration experiment

At the end of the experiment, you will have a set of readings for the masses and the accelerations. You should find that the product of the corresponding masses and accelerations are all equal - this value is the downward force of gravity due to the masses on the end of the string. You can check your result by using the formula stated in the first section,

$$W=mg.$$

There are several key points to consider in this experiment so that you can obtain the most accurate results:

• There will be some friction between the cart and the table which will slow the cart down. This can be partly prevented by using a smooth surface.
• There will be some friction between the pulley and the string. This effect can be reduced by using a new pulley and a string that is smooth so that it has no tears in it.
• There will also be frictional forces due to air resistance acting on the cart and the hanging mass.
• All the masses used, including the cart, must be accurately measured or the calculations of the force will be inaccurate.
• Check if there are any anomalous results. It is sometimes easy to note down the wrong number or use the wrong number of masses to load the cart.

When carrying out this experiment, you should also pay attention to the following safety hazards:

• Place something soft, such as a pillow, underneath the masses so that they do not damage the floor.
• Check that the mains cable and plug connected to the datalogger are not broken to avoid electrical faults.

Mass and acceleration graph

We can use our results for the masses and accelerations to plot a graph to show the validity of Newton's second law. The formula for Newton's second law of motion is

$$F=ma.$$

In this experiment, we measured the mass and the acceleration, so we want to plot these against each other to show that the force remains constant - as the mass of the cart increases, the acceleration decreases enough so that their product is the same force. If we rearrange the formula to

$$a=\frac Fm,$$

then we can see from this equation that if we use our results to plot the points on a graph of \( a \) against \( \frac 1m \), then the gradient of the line of best fit will be \( F \). If the gradient is constant then we will have shown that these masses and accelerations obey Newton's second law and hopefully, the gradient \( F \) will be equal to the weight of the hanging masses.

Fig. 5 - An example of a graph that might be obtained by performing this experiment.

This experiment is a relatively simple way to show the validity of Newton's second law. There are some sources of error (which were mentioned above) that might cause the points on the graph to deviate from the expected straight line, as shown in Fig. 5. However, the points should still roughly follow the overall relationship given by Newton's second law. You can perform several different experiments to test Newton's second law. For example, if you measured the force acting on an object of unknown mass and measured its acceleration for each force, you could plot a graph of force against acceleration to find the mass of the object as the gradient.