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A block with a mass m = 3.50\ \mathrm{kg} sits on a surface with a coefficient of static friction...

Question:

A block with a mass {eq}m = 3.50\ \mathrm{kg} {/eq} sits on a surface with a coefficient of static friction {eq}\mu s = 0.520 {/eq}. A spring with spring constant {eq}k = 75.0\ \mathrm{N/m} {/eq} is hooked onto the block, and is slowly pulled horizontally, gradually stretching the spring.

a. When the spring is stretched {eq}10.0\ \mathrm{cm} {/eq}, the block is still stationary. What is the friction force felt by the block when the spring is stretched {eq}10.0\ \mathrm{cm} {/eq}?

b. How much is the spring stretched when the block just begins to move?

Hooke's Law For Springs:

Hooke's law, named after Robert Hooke, relates the stretch {eq}e {/eq} of an elastic spring of spring constant {eq}k {/eq} and the force {eq}F {/eq} it applies on an object attached to it. The mathematical form of the law is:

{eq}F = ke {/eq}

Answer and Explanation: 1

We are given the force constant as {eq}k = 75.0 \ \text{N/m} {/eq}.

(a) The stretch on the spring is:

{eq}e = 10.0 \ \text{cm} = 0.1 \ \text{m} {/eq}

Invoking Hooke's law, we write:

{eq}\begin{align} F &= ke \\ &= (75.0 \ \text{N/m})(0.1 \ \text{m}) \\ &= 7.5 \ \text{N} \end{align} {/eq}

This is also the frictional force felt by the block since it is stationary..

(b)

The coefficient of static (maximum) friction is given as {eq}\mu_s = 0.520 {/eq}, and the weight of the block can be calculated as:

{eq}\begin{align} W &= mg \\ &= (3.50 \ \text{kg})(9.8 \ \text{m/s}^2) \\ &= 34.3 \ \text{N} \end{align} {/eq}.

Hence, the normal reaction on this block from the surface is:

{eq}R = 34.3 \ \text{N} {/eq}.

The maximum frictional force, achieved just before motion occurs, is:

{eq}\begin{align} F &= \mu R \\ &= (0.520)(34.3 \ \text{N}) \\ &= 17.84 \ \text{N} \end{align} {/eq} .

The corresponding stretch is, then,

{eq}\displaystyle \begin{align} e &= \frac{F}{k} \\ &= \frac{17.84 \ \text{N}}{75.0 \ \text{N/m}} \\ &= 0.238 \ \text{m} \end{align} {/eq}


Learn more about this topic:

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Hooke's Law & the Spring Constant: Definition & Equation

from

Chapter 4 / Lesson 19
201K

After watching this video, you will be able to explain what Hooke's Law is and use the equation for Hooke's Law to solve problems. A short quiz will follow.


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