Math / Algebra

QuestionFIGURE 5-38 Problem 26. (II) A child slides down a slide with a $28^{\circ}$ incline, and at the bottom her speed is precisely half what it would have been if the slide had been frictionless. Calculate the coefficient of kinetic friction between the slide and the child.

Studdy Solution

STEP 1

1. The slide has an incline angle of $28^{\circ}$ .
2. The child's speed at the bottom is half of what it would be if there were no friction.
3. We need to calculate the coefficient of kinetic friction ( $\mu_k$ ) between the slide and the child.

STEP 2

1. Analyze the energy conservation for the frictionless slide.
2. Analyze the energy conservation for the slide with friction.
3. Relate the two scenarios to find the coefficient of kinetic friction.

STEP 3

Analyze the energy conservation for the frictionless slide.
- In the frictionless scenario, all potential energy converts to kinetic energy. - Let $h$ be the height of the slide, $m$ the mass of the child, and $v_f$ the final speed without friction. - Potential energy at the top: $mgh$ . - Kinetic energy at the bottom: $\frac{1}{2}mv_f^2$ . - By conservation of energy: $mgh = \frac{1}{2}mv_f^2$ .

STEP 4

Analyze the energy conservation for the slide with friction.
- In the scenario with friction, some energy is lost to friction. - Let $v$ be the final speed with friction. - Given that $v = \frac{1}{2}v_f$ . - Energy equation: $mgh = \frac{1}{2}mv^2 + \text{work done by friction}$ . - Work done by friction: $f_k \cdot d = \mu_k \cdot mg \cdot \cos(\theta) \cdot d$ , where $d$ is the slide length.

STEP 5

Relate the two scenarios to find the coefficient of kinetic friction.
- From STEP_1, $v_f^2 = 2gh$ . - From STEP_2, $v^2 = \frac{1}{4}v_f^2 = \frac{1}{4}(2gh) = \frac{1}{2}gh$ . - Substitute into energy equation: $mgh = \frac{1}{2}m(\frac{1}{2}gh) + \mu_k \cdot mg \cdot \cos(\theta) \cdot d$ . - Simplify: $mgh - \frac{1}{4}mgh = \mu_k \cdot mg \cdot \cos(\theta) \cdot d$ . - $\frac{3}{4}mgh = \mu_k \cdot mg \cdot \cos(\theta) \cdot d$ . - Cancel $mg$ and solve for $\mu_k$ : $\mu_k = \frac{3h}{4d \cos(\theta)}$ .

STEP 6

Express $h$ in terms of $d$ and $\theta$ .
- From geometry, $h = d \cdot \sin(\theta)$ . - Substitute: $\mu_k = \frac{3d \cdot \sin(\theta)}{4d \cdot \cos(\theta)}$ . - Simplify: $\mu_k = \frac{3 \tan(\theta)}{4}$ .

STEP 7

Calculate $\mu_k$ using $\theta = 28^{\circ}$ .
- Calculate $\tan(28^{\circ})$ . - $\mu_k = \frac{3 \tan(28^{\circ})}{4}$ . - Use a calculator to find $\tan(28^{\circ}) \approx 0.5317$ . - $\mu_k = \frac{3 \times 0.5317}{4} \approx 0.3988$ .
The coefficient of kinetic friction is approximately:
$\boxed{0.3988}$

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QuestionFIGURE 5-38 Problem 26. (II) A child slides down a slide with a $28^{\circ}$ incline, and at the bottom her speed is precisely half what it would have been if the slide had been frictionless. Calculate the coefficient of kinetic friction between the slide and the child.

Studdy Solution

STEP 1

1. The slide has an incline angle of $28^{\circ}$ .
2. The child's speed at the bottom is half of what it would be if there were no friction.
3. We need to calculate the coefficient of kinetic friction ( $\mu_k$ ) between the slide and the child.

STEP 2

1. Analyze the energy conservation for the frictionless slide.
2. Analyze the energy conservation for the slide with friction.
3. Relate the two scenarios to find the coefficient of kinetic friction.

STEP 3

STEP 4

STEP 5

STEP 6

Express $h$ in terms of $d$ and $\theta$ .
- From geometry, $h = d \cdot \sin(\theta)$ . - Substitute: $\mu_k = \frac{3d \cdot \sin(\theta)}{4d \cdot \cos(\theta)}$ . - Simplify: $\mu_k = \frac{3 \tan(\theta)}{4}$ .

STEP 7

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QuestionFIGURE 5-38 Problem 26. (II) A child slides down a slide with a 28∘28^{\circ}28∘ incline, and at the bottom her speed is precisely half what it would have been if the slide had been frictionless. Calculate the coefficient of kinetic friction between the slide and the child.

Studdy Solution

STEP 1

1. The slide has an incline angle of 28∘28^{\circ}28∘.2. The child's speed at the bottom is half of what it would be if there were no friction.3. We need to calculate the coefficient of kinetic friction (μk\mu_kμk​) between the slide and the child.

STEP 2

1. Analyze the energy conservation for the frictionless slide.2. Analyze the energy conservation for the slide with friction.3. Relate the two scenarios to find the coefficient of kinetic friction.

STEP 3

STEP 4

STEP 5

STEP 6

STEP 7

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionFIGURE 5-38 Problem 26. (II) A child slides down a slide with a $28^{\circ}$ incline, and at the bottom her speed is precisely half what it would have been if the slide had been frictionless. Calculate the coefficient of kinetic friction between the slide and the child.

1. The slide has an incline angle of $28^{\circ}$ .
2. The child's speed at the bottom is half of what it would be if there were no friction.
3. We need to calculate the coefficient of kinetic friction ( $\mu_k$ ) between the slide and the child.

1. Analyze the energy conservation for the frictionless slide.
2. Analyze the energy conservation for the slide with friction.
3. Relate the two scenarios to find the coefficient of kinetic friction.