Find the Coefficient of Kinetic Friction

In summary, the authors suggest that the crate has a mass of 180 kg, and that the washer has a mass of 55 kg. Newton's first and second laws are used to derive the expression for the washer's mass. The washer has a mass of 68 kg, which is the same as the mass of the crate.
  • #1
Argonaut
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Homework Statement
A steel washer is suspended inside an empty shipping crate from a light string attached to the top of the crate. The crate slides down a long ramp that is inclined at an angle of 37° above the horizontal. The crate has mass 180 kg. You are sitting inside the crate(with a flashlight); your mass is 55 kg. As the crate is sliding down the ramp, you find the washer is at rest with respect to the crate when the string makes an angle of 68° with the top of the crate. What is the coefficient of kinetic friction between the ramp and the crate?
Relevant Equations
Newton's First and Second Laws
Young & Freedman 13th ed, Exercise 7.81

unnamed (1).jpg


Starting with the crate, here is its free-body diagram:
unnamed.jpg

In accordance with Newton's First Law:

$$ \Sigma F_y = 0 = n+(-w_c \cos{\alpha}) $$

Thus ## n = w_c \cos{\alpha} ##.

And according to Newton's Second Law:

$$ \Sigma F_x = m_c a_x = w_c \sin{\alpha} + (-f) $$

Utilising the fact that ## f = \mu_k n ## and substituting n with the expression above, we get:

$$
\begin{align*}
m_c a_x &= w_c \sin{\alpha} - \mu_k n \\
&= w_c \sin{\alpha} - \mu_k w_c \cos{\alpha} \\
&= m_c g (\sin{\alpha} - \mu_k \cos{\alpha})
\end{align*}
$$

Thus ## \mu_k = \frac{\sin{\alpha} - a_x/g}{\cos{\alpha}} ##.

##a_x## is unknown. Switching to the washer, it's free-body diagram:

unnamed (2).jpg


According to Newton's First Law:

$$ \Sigma F_y = 0 = T \sin{\beta} - w_w \cos{\alpha} $$

Thus we have ## T = w_w \frac{\cos{\alpha}}{\sin{\beta}} ##.

According to Newton's Second Law:

$$ \Sigma F_x = m_w a_x = w_w \sin{\alpha} -T \cos{\beta} $$

Substituting T with the expression above and rearranging to express ## a_x ##, we get:

$$
\begin{align*}
m_w a_x &= w_w \sin{\alpha} -T \cos{\beta} \\
&= w_w \sin{\alpha} - w_w \frac{\cos{\alpha}}{\sin{\beta}} \cos{\beta}
\end{align*}
$$

Thus ## a_x = g( \sin{\alpha} - \frac{\cos{\alpha}}{\sin{\beta}} \cos{\beta}) ##

Plugging in ##a_x##to the expression for ## \mu_k##, we get:

$$
\begin{align*}
\mu_k &= \sin{\alpha}( \sin{\alpha} - \frac{\cos{\alpha}}{\sin{\beta}} \cos{\beta})/\cos{\alpha} \\
&= \frac{\cos{\beta}}{\sin{\alpha}} \\
&= \frac{\cos{68°}}{\sin{37°}} \\
&\approx 0.62
\end{align*}
$$

The book solution says it's 0.40. Where did I make a mistake?
 
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  • #2
Argonaut said:
Plugging in ##a_x##to the expression for ## \mu_k##, we get:
Check your algebra for 'plugging in' and what follows.

Also, you have used the mass and weight of the crate alone in various equations – but you should have used totals (crate + person). However, since ##m## ans ##w## cancel-out in the various equations, it shouldn’t affect the answer.
 
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  • #3
Argonaut said:
$$
\begin{align*}
\mu_k &= \sin{\alpha}-( \sin{\alpha} - \frac{\cos{\alpha}}{\sin{\beta}} \cos{\beta})/\cos{\alpha} \\
&= \frac{\cos{\beta}}{\sin{\alpha}} \\
&= \frac{\cos{68°}}{\sin{37°}} \\
&\approx 0.62
\end{align*}
$$

The book solution says it's 0.40. Where did I make a mistake?

Don't mind me - I've found it. I made a mistake while doing the algebraic manipulation at the end. Very silly. (And there was a mistake in my LaTeX too).
The correct seqence:

$$
\begin{align*}
\mu_k &= (\sin{\alpha}-( \sin{\alpha} - \frac{\cos{\alpha}}{\sin{\beta}} \cos{\beta}))/\cos{\alpha} \\
&= \frac{\cos{\beta}}{\sin{\beta}} \\
&= \frac{\cos{68°}}{\sin{68°}} \\
&\approx 0.40
\end{align*}
$$

Lesson: Don't skip steps - better to write everything out if you are bad at mental arithmetic.
 
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  • #4
Steve4Physics said:
Check your algebra for 'plugging in' and what follows.

Also, you have used the mass and weight of the crate alone in various equations – but you should have used totals (crate + person). However, since ##m## ans ##w## cancel-out in the various equations, it shouldn’t affect the answer.
Thanks!

About the mass of the crate, the exercise says "The crate has mass 180 kg". I assumed it meant the crate including its content, so the person and the washing machine. Do you think they meant the empty crate? Even if as you say, it's moot as the mass cancels out.
 
  • #5
Argonaut said:
Thanks!

About the mass of the crate, the exercise says "The crate has mass 180 kg". I assumed it meant the crate including its content, so the person and the washing machine. Do you think they meant the empty crate? Even if as you say, it's moot as the mass cancels out.
The question gives separate values of mass for the crate and the person (180kg and 55kg). I interpret that as a total mass of 235kg. For example, if asked to calculate the frictional force, I would use 235kg in the calculation.

Note: A ‘washer’ in the context of this question is a small metal disc with a hole in the middle. E.g. see https://www.ssflatwashers.com/china...e_diameter_washer_clean_surface-11725011.html

You would not want to be trapped inside a crate, sliding down a hill, with a washing machine suspended by a 'light string' hanging over you! (Now I understand your Post #1 diagram!)
 
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  • #6
Steve4Physics said:
The question gives separate values of mass for the crate and the person (180kg and 55kg). I interpret that as a total mass of 235kg. For example, if asked to calculate the frictional force, I would use 235kg in the calculation.

Note: A ‘washer’ in the context of this question is a small metal disc with a hole in the middle. E.g. see https://www.ssflatwashers.com/china...e_diameter_washer_clean_surface-11725011.html

You would not want to be trapped inside a crate, sliding down a hill, with a washing machine suspended by a 'light string' hanging over you! (Now I understand your Post #1 diagram!)
Ah hahaha, I love it! It did seem like a quirky setup :DD

Thanks for the feedback again.
 
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Related to Find the Coefficient of Kinetic Friction

1. What is the coefficient of kinetic friction?

The coefficient of kinetic friction is a measure of the resistance to motion between two surfaces in contact when one of the surfaces is in motion. It is denoted by the symbol "μk" and is a dimensionless quantity.

2. How is the coefficient of kinetic friction calculated?

The coefficient of kinetic friction can be calculated by dividing the force of kinetic friction by the normal force between the two surfaces. Mathematically, it can be represented as μk = Fk/Fn, where Fk is the force of kinetic friction and Fn is the normal force.

3. What factors affect the coefficient of kinetic friction?

The coefficient of kinetic friction is affected by several factors, including the nature of the surfaces in contact, the roughness of the surfaces, and the presence of any lubricants or contaminants between the surfaces. It also depends on the magnitude of the applied force and the speed of the moving surface.

4. Why is the coefficient of kinetic friction important?

The coefficient of kinetic friction is an important concept in physics and engineering as it helps in understanding the behavior of objects in motion. It is also crucial in determining the amount of force needed to keep an object moving at a constant speed on a given surface.

5. How is the coefficient of kinetic friction used in real-world applications?

The coefficient of kinetic friction is used in various real-world applications, such as designing brake systems for vehicles, determining the amount of force needed to push or pull an object, and predicting the motion of objects on different surfaces. It is also used in sports, such as determining the speed of a hockey puck on an ice rink or the distance a baseball will travel on a grass field.

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