Ch. 05 - Using Newton's Laws: Friction, Circular Motion, Drag Forces

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 05 - Using Newton's Laws: Friction, Circular Motion, Drag Forces

Problem 80

Chapter 5, Problem 80

A coffee cup on the horizontal dashboard of a car slides forward when the driver decelerates from 45 km/h to rest in 3.5 s or less, but not if she decelerates in a longer time. What is the coefficient of static friction between the cup and the dash? Assume the road and the dashboard are level (horizontal).

Verified step by step guidance

Convert the initial velocity of the car from km/h to m/s. Use the conversion factor: $1 \text{ km/h} = \frac{1}{3.6} \text{ m/s}$. Thus, $v_i = 45 \text{ km/h} = \frac{45}{3.6} \text{ m/s}$.

Determine the acceleration required to bring the car to rest in 3.5 seconds. Use the kinematic equation $a = \frac{v_f - v_i}{t}$, where $v_f = 0 \text{ m/s}$ (final velocity), $v_i$ is the initial velocity (calculated in step 1), and $t = 3.5 \text{ s}$.

Recognize that the coffee cup slides forward when the deceleration exceeds the maximum static friction force. The maximum static friction force is given by $f_s = \mu_s N$, where $\mu_s$ is the coefficient of static friction and $N$ is the normal force. On a horizontal surface, $N = mg$, where $m$ is the mass of the cup and $g$ is the acceleration due to gravity ($9.8 \text{ m/s}^2$).

The force causing the deceleration of the car is $F = ma$, where $a$ is the acceleration (calculated in step 2). For the cup to remain stationary relative to the dashboard, the static friction force must equal or exceed this force. Thus, $\mu_s mg \geq ma$.

Simplify the inequality $\mu_s mg \geq ma$ to solve for $\mu_s$. Cancel $m$ from both sides (assuming $m \neq 0$), resulting in $\mu_s \geq \frac{a}{g}$. Substitute the value of $a$ (from step 2) and $g = 9.8 \text{ m/s}^2$ to find $\mu_s$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Newton's First Law of Motion

Newton's First Law states that an object at rest will remain at rest, and an object in motion will remain in motion at a constant velocity unless acted upon by a net external force. In this scenario, the coffee cup slides forward due to the deceleration of the car, illustrating that the cup's inertia resists the change in motion.

Friction

Friction is the force that opposes the relative motion of two surfaces in contact. The coefficient of static friction quantifies the maximum frictional force that can act before an object begins to slide. In this case, it determines whether the cup remains stationary on the dashboard during deceleration.

Deceleration and Acceleration

Deceleration is a form of acceleration that results in a decrease in velocity. The rate of deceleration affects the forces acting on the coffee cup. If the deceleration exceeds the maximum static friction force, the cup will slide, allowing us to calculate the coefficient of static friction based on the car's deceleration.

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