Question

You are going sledding with your friends, sliding down a snowy hill. Friction can't be ignored. Riding solo on your sled, you have a certain acceleration. Would the acceleration change if you let a friend ride with you, increasing the mass?

No, increasing the mass does increase the net force on the system, but it also increases the inertia. \(a=\frac{m}{F_{n e}} .\) Since both the net force and mass are increased they still Cancel, leaving the acceleration the same.

Yes, increasing the mass does increase the inertia. \(a=\frac{F_{\text {net }}}{m} .\) Since the mass is increased, the acceleration decreases.

No, increasing the mass does increase the net force on the system, but it also increases the inertia. \(a=\frac{F_{\text {net }}}{m} .\) Since both the net force and mass are increased they still cancel, leaving the acceleration the same.

Yes, increasing the mass does increase the net force on the system. \(a=\frac{F_{\text {net }}}{m} .\) Since the net force is increased, the acceleration increases too

Answer 1

Concept and reason

The concept used to solve this problem is frictional force and Newton’ second law of motion.

Use Newton’s law of motion to resolve the components of forces acting on the person over a frictionless plane and find the relationship between mass and acceleration to calculate the change in acceleration on increasing mass.

Fundamentals

The free body diagram below shows the force acting on the person.

Using Newton’s law of motion, the net force of the person is expressed as

${F_{net}} = ma$

Here, ${F_{net}}$ is the net force, m is the mass of the person, and a is the acceleration of the person.

Gravitational force acting vertically downwards on the person splits into vertical and horizontal when inclined.

The vertical component of force acting due to gravity is,

$N = mg\cos \theta$

Here, N is the normal force between the person and an inclined plane and $\theta$ is the angle inclination.

The expression for the frictional force is,

${f_k} = \mu N$

Here, $\mu$ is the coefficient of friction.

Substitute $mg\cos \theta$ for N.

${f_k} = \mu mg\cos \theta$

The horizontal component of force acting due to gravity is,

${F_{net}} = mg\sin \theta - {f_k}$

Here, ${f_k}$ is the frictional force.

The horizontal component of force acting due to gravity is,

${F_{net}} = mg\sin \theta - {f_k}$

Substitute $\mu mg\cos \theta$ for ${f_k}$ and $ma$ for ${F_{net}}$ .

$\begin{array}{c}\\{F_{net}} = mg\sin \theta - \mu mg\cos \theta \\\\ma = mg\left( {\sin \theta - \mu \cos \theta } \right)\\\end{array}$

Rearrange the equation to get the acceleration of the first person,

$a = g\left( {\sin \theta - \mu \cos \theta } \right)$

The net force of the person after the addition of a second person is,

${F_{net}} = \left( {m + M} \right)a$

Here, M is the mass of the second person.

The vertical component of force acting due to gravity in addition with the second person is,

$N = \left( {m + M} \right)g\cos \theta$

The expression for the frictional force in addition of the second person is,

${f_k} = \mu N$

Substitute $\left( {m + M} \right)g\cos \theta$ for N.

${f_k} = \mu \left( {m + M} \right)g\cos \theta$

The horizontal component of force acting due to gravity is,

${F_{net}} = \left( {m + M} \right)g\sin \theta - {f_k}$

Substitute $\mu \left( {m + M} \right)g\cos \theta$ for ${f_k}$ and $\left( {m + M} \right)a$ for ${F_{net}}$ .

$\left( {m + M} \right)a = \left( {m + M} \right)g\sin \theta - \mu \left( {m + M} \right)g\cos \theta$

Rearrange the equation to get the acceleration of the second person,

$a = g\left( {\sin \theta - \mu \cos \theta } \right)$

The acceleration of the first person is equal to the acceleration of the second person. This states that the acceleration remains the same on increasing mass.

The following are incorrect options to calculate the acceleration of the person on increasing mass.

• Yes, increasing the mass does increase the inertia. $a = \frac{{{F_{net}}}}{m}$ . Since the mass is increased, the acceleration decreases.

• Yes, increasing the mass does increase the net force on the system. $a = \frac{{{F_{net}}}}{m}$ . Since the net force is increased, the acceleration increases too.

• No, increasing the mass does increase the net force on the system, but it also increases the inertia. $a = \frac{m}{{{F_{net}}}}$ . Since both the net force and mass are increased, they still get cancelled, leaving the acceleration the same.

The net force acting on the person is,

${F_{net}} = ma$

So by increasing the mass, the net force also keeps increasing which keeps the acceleration constant.

$a = \frac{{{F_{net}}}}{m}$

The acceleration of the person is,

$a = g\left( {\sin \theta - \mu \cos \theta } \right)$

Ans:

No, increasing the mass does increase the net force on the system, but it also increases the inertia. $a = \frac{{{F_{net}}}}{m}$ . Since both the net force and mass are increased, they still get cancelled, leaving the acceleration the same.

Answer 2