## A girl pulls a sledge on some snowy pavement

A girl pulls a loaded sledge with mass m = 20 kg on some horizontal snowy pavement. The velocity of the sledge is constant. The coefficient of kinetic friction μk between the runner and the pavement is 0.1 and the angle φ between the cord and the pavement is 30°. Find:

1) The acceleration of the sledge

2) The magnitude of the tension force by which the girl affects the sledge

3) The magnitude of the normal force by which the sledge affects the pavement

4) The magnitude of the friction force that affects the sledge

• #### Hint 1 – Acceleration of the sledge, the force that affects the sledge

What does it mean for the acceleration (and also the resulting force) when the velocity is constant? Draw a picture and mark all the forces that affect the sledge. Use Newton’s second law $$(\Sigma\vec{F} = m\vec{a})$$ and write the vector motion equation for the sledge.

• #### Hint 2 – The scalar motion equations

The motion equations need to be written down in scalar form. Choose the appropriate coordinate system, find the components of the forces in the corresponding directions and rewrite the motion equations in these directions.

• #### Hint 3 – The magnitudes of the tension force and the friction force

Think about what the friction force depends on and how it could be expressed.

• #### Hint 4 – The magnitude of the normal force

The force by which the sledge affects the pavement is according to Newton’s third law of the same magnitude as the force by which the pavement affects the sledge. The magnitude of the force N (by which the pavement affects the sledge) is given by equation (5): $N = mg - T\sin\varphi\tag{5}$ It just remains to substitute.

• #### Hint 5 – The magnitude of the friction force

The magnitude of the friction force Ft is given by equation (4): $F_{t} = \mu_{k} N \tag{4}$ It just remains to substitute.

• #### COMPLETE SOLUTION OF POINT 1 – Acceleration of the sledge

If the velocity of the sledge is constant, its velocity doesn’t change with time and thus the acceleration of the sledge equals zero $$(\vec{a} = 0).$$

Answer: The acceleration of the sledge a equals zero.

• #### COMPLETE SOLUTION OF POINT 2 – The magnitude of the tension force

Forces that affect the sledge:

$$\vec{F}_{G}$$...the weight

$$\vec{N}$$...force by which the pavement affects the sledge vertically (the normal force)

$$\vec{T}$$...desired tension force by which the girl pulls the sledge

$$\vec{F}_{t}$$...friction force which opposes the sledge’s motion

The vector motion equation for the sledge:

(since $$\vec{F}_{net} = m \vec{a}$$, and $$\vec{a} = 0$$, the vector sum of the forces must be zero)

$\vec{F}_{G}+\vec{N}+\vec{T}+\vec{F}_{t}=m\vec{a}=0\tag{1}$

m…mass of the sledge

$$\vec{a}$$…acceleration of the sledge

Because the velocity $$\vec{v}$$ is constant, the acceleration $$\vec{a}$$ equals zero and the resultant force that affects the sledge is also equal to zero.

Choose the appropriate coordinate system and rewrite the motion equations in scalar form. Choose the x-axis horizontally in the direction of the motion and choose the y-axis vertically – as shown in the picture.

The force equation in the x-direction:

$\sum^\ F_x = T\cos\varphi - F_{t} = 0 \tag{2}$

The force equation in the y-direction:

$\sum^\ F_y = N + T\sin\varphi - F_G = 0 \tag{3}$

The friction force is directly proportional to the force by which the sledge affects the pavement. This force is according to Newton’s third law of the same magnitude as the force by which the pavement affects the sledge.

The friction force Ft can be expressed as:

$F_t = \mu_k N \tag{4}$

The force N (by which the pavement affects the sledge) can be expressed using equation (3):

$N = mg - T\sin\varphi\tag{5}$

Equations (4) and (5) give:

$F_{t} = \mu_{k}(mg - T\sin\varphi)\tag{6}$

Substitute equation (6) into equation (2), and then write the projection to the x-axis as:

$T\cos\varphi - \mu_{k}(mg - T\sin\varphi) = 0\tag{7}$

Express the tension force from the equation (7):

$T = \frac{mg\mu_{k}}{\cos\varphi + \mu_{k}\sin\varphi}\tag{8}$

For the given numerical values we get from (8):

$T = \frac{20{\cdot}9.81{\cdot}0.1}{\cos 30^{\circ} + 0.1\cdot\sin30^{\circ}}\,\textrm{N} =\frac{19.62}{\frac{\sqrt{2}}{2} + 0.1{\cdot}0.5}\,\textrm{N}$ $T=21.4\,\textrm{N}$

Answer: The magnitude of the tension force T by which the girl pulls the sledge is 21.4 N.

• #### COMPLETE SOLUTION OF POINT 3 – The magnitude of the normal force

The force by which the sledge affects the pavement is according to Newton’s third law of the same magnitude as the force by which the pavement affects the sledge. The magnitude of the force N (by which the pavement affects the sledge) is given by equation (5):

$N = mg - T\sin\varphi\tag{5}$

Substitute for T from equation (8), and you get the expression:

$N = mg - \frac{mg\mu_{k}}{\cos\varphi + \mu_{k}\sin\varphi}\sin\varphi = mg (1 - \frac{\mu_{k}\sin\varphi}{\cos\varphi + \mu_{k}\sin\varphi})=$ $\,\,\,\,\,\,\,\,= mg(\frac{\cos\varphi}{\cos\varphi + \mu_{k}\sin\varphi}) = mg(\frac{1}{1 + \mu_{k}\tan\varphi})\tag{9}$

For the given numerical values we get from (9):

$N = 20{\cdot}9.81(\frac{1}{1 + 0.1\tan30^{\circ}})\,\textrm{N}= 196.2(\frac{1}{1 + 0.1\frac{\sqrt{3}}{3}})\,\textrm{N}$ $N=185.5\,\textrm{N}$

Answer: The magnitude of the normal force N (by which the sledge affects the pavement) is 185.5 N.

• #### COMPLETE SOLUTION OF POINT 4 – The magnitude of the friction force

The magnitude of the friction force Ft is given by equation (4):

$F_{t} = \mu_{k} N \tag{3}$

If you substitute for N in equation (4) using equation (9), then:

$F_{t} = mg(\frac{1}{1 + \mu_{k}\tan\varphi})\mu_{k}\tag{9}$

For the given numerical values we get from (10):

$F_t= 20{\cdot}9.81(\frac{1}{1 + 0.1\cdot\tan30^{\circ}})0.1\,\textrm{N}= 19.62(\frac{1}{1 + 0.1\frac{\sqrt{3}}{3}})\,\textrm{N}$ $F_t=18.6\,\textrm{N}$

Answer: The magnitude of the friction force Ft that affects the sledge is 18.6 N.

1) The acceleration of the sledge a equals zero.

2) The magnitude of the tension force T (by which the girl pulls the sledge) is

$T = \frac{mg\mu_{k}}{\cos\varphi + \mu_{k}\sin\varphi}=21.4\,\textrm{N}.$

3) The magnitude of the normal force N (by which the sledge affects the pavement) is

$N = mg(\frac{1}{1 + \mu_{k}\tan\varphi})=185.5\,\textrm{N}.$

4) The magnitude of the friction force Ft that affects the sledge is

$F_{t} = mg(\frac{1}{1 + \mu_{k}\tan\varphi})\mu_{k}=18.6\,\textrm{N}.$