1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
tino4ka555 [31]
3 years ago
6

A 1.10-kg object slides to the right on a surface having a coefficient of kinetic friction 0.250 (Figure a). The object has a sp

eed of vi = 2.60 m/s when it makes contact with a light spring (Figure b) that has a force constant of 50.0 N/m. The object comes to rest after the spring has been compressed a distance d (Figure c). The object is then forced toward the left by the spring (Figure d) and continues to move in that direction beyond the spring's unstretched position. Finally, the object comes to rest a distance D to the left of the unstretched spring (Figure e).
The right end of a horizontal spring labeled k is attached to a wall. Five images show five configurations as a block labeled m approaches, compresses, and then moves away from the spring.
In figure a, the block is to the left of the spring, and an arrow above the block points to the right.
In figure b, the block is just touching the uncompressed spring, and an arrow labeled vector vi above the block points to the right.
In figure c, the block has compressed the spring by a distance d, and a label indicates vector vf = 0.
In figure d, the block is just touching the uncompressed spring, and an arrow labeled vector v above the block points to the left.
In figure e, the block is a distance D away from the spring, and a label indicates vector v = 0.
(a)
Find the distance of compression d (in m).
m
(b)
Find the speed v (in m/s) at the unstretched position when the object is moving to the left (Figure d).
m/s
(c)
Find the distance D (in m) where the object comes to rest.
m
(d)
What If? If the object becomes attached securely to the end of the spring when it makes contact, what is the new value of the distance D (in m) at which the object will come to rest after moving to the left?
m

Physics
1 answer:
cluponka [151]3 years ago
7 0

Answer:

(a) Approximately 0.335\; \rm m.

(b) Approximately 1.86\; \rm m\cdot s^{-1}.

(c) Approximately 0.707\; \rm m.

(d) Approximately 0.228\; \rm m.

Explanation:

  • v_i denotes the velocity of the object in the first diagram right before it came into contact with the spring.
  • Let m denote the mass of the block.
  • Let \mu denote the constant of kinetic friction between the object and the surface.
  • Let g denote the constant of gravitational acceleration.
  • Let k denote the spring constant of this spring.
<h3>(a)</h3>

Consider the conversion of energy in this object-spring system.

First diagram: Right before the object came into contact with the spring, the object carries kinetic energy \displaystyle \frac{1}{2}\, m \cdot {v_{i}}^2.

Second diagram: As the object moves towards the position in the third diagram, the spring gains elastic potential energy. At the same time, the object loses energy due to friction.

Third diagram: After the velocity of the object becomes zero, it has moved a distance of D and compressed the spring by the same distance.

  • Energy lost to friction: \underbrace{(\mu \cdot m \cdot g)}_{\text{friction}} \cdot D.
  • Elastic potential energy that the spring has gained: \displaystyle \frac{1}{2}\,k\, D^2.

The sum of these two energies should match the initial kinetic energy of the object (before it comes into contact with the spring.) That is:

\displaystyle \frac{1}{2}\, m \cdot {v_{i}}^{2} = (\mu\cdot m \cdot g) \cdot D + \frac{1}{2}\, k \cdot D^2.

Assume that g = 9.81\; \rm m \cdot s^{-2}. In the equation above, all symbols other than D have known values:

  • m =1.10\; \rm kg.
  • v_i = 2.60\; \rm m \cdot s^{-1}.
  • \mu = 0.250.
  • g = 9.81\; \rm m \cdot s^{-2}.
  • k = 50.0\; \rm N \cdot m^{-1}.

Substitute in the known values to obtain an equation for D (where the unit of D\! is m.)

3.178 = 2.69775\, D + 25\, D^2.

2.69775\, D + 25\, D^2 + 3.178 = 0.

Simplify and solve for D. Note that D > 0 because the energy lost to friction should be greater than zero.

D \approx 0.335\; \rm m.

<h3>(b)</h3>

The energy of the object-spring system in the third diagram is the same as the elastic potential energy of the spring:

\displaystyle \frac{1}{2}\,k\, D^2 \approx 2.81\; \rm J.

As the object moves to the left, part of that energy will be lost to friction:

(\mu \cdot m \cdot g) \, D \approx 0.905\; \rm J.

The rest will become the kinetic energy of that block by the time the block reaches the position in the fourth diagram:

2.81\; \rm J - 0.905\; \rm J \approx 1.91\; \rm J.

Calculate the velocity corresponding to that kinetic energy:

\displaystyle v =\sqrt{\frac{2\, (\text{Kinetic Energy})}{m}} \approx 1.86\; \rm m \cdot s^{-1}.

<h3>(c)</h3>

As the object moves from the position in the fourth diagram to the position in the fifth, all its kinetic energy (1.91\; \rm J) would be lost to friction.

How far would the object need to move on the surface to lose that much energy to friction? Again, the size of the friction force is \mu \cdot m \cdot g.

\displaystyle (\text{Distance Travelled}) = \frac{\text{(Work Done by friction)}}{\text{(Size of the Friction Force)}} \approx0.707\; \rm m.

<h3>(d)</h3>

Similar to (a), solving (d) involves another quadratic equation about D.

Left-hand side of the equation: kinetic energy of the object (as in the fourth diagram,) 1.91\; \rm J.

Right-hand side of the equation: energy lost to friction, plus the gain in the elastic potential energy of the spring.

\displaystyle {1.91\; \rm J} \approx (\mu\cdot m \cdot g) \cdot D + \frac{1}{2}\, k \cdot D^2.

25\, D^2 + 2.69775\, D - 1.90811\approx 0.

Again, D > 0 because the energy lost to friction is greater than zero.

D \approx 0.228\; \rm m.

You might be interested in
All atoms of an element have the same number of__But within the same element there may exist atoms that have__numbers of__.Atoms
Elza [17]

Answer:

isotopes

Explanation:

3 0
3 years ago
How will the positions of the police car and the truck compare when they have the same speed and why?
sdas [7]

Answer:

Two cars of equal weight and braking ability are travelling along the same road but combined with other factors it could mean the difference between life.

7 0
3 years ago
Light of wavelength 400 nm is incident on a single slit of width 15 microns. If a screen is placed 2.5 m from the slit. How far
olganol [36]

Answer:

0.0667 m

Explanation:

λ = wavelength of light = 400 nm = 400 x 10⁻⁹ m

D = screen distance = 2.5 m

d = slit width = 15 x 10⁻⁶ m

n = order = 1

θ = angle = ?

Using the equation

d Sinθ = n λ

(15 x 10⁻⁶) Sinθ = (1) (400 x 10⁻⁹)

Sinθ = 26.67 x 10⁻³

y = position of first minimum

Using the equation for small angles

tanθ = Sinθ = y/D

26.67 x 10⁻³ = y/2.5

y = 0.0667 m

5 0
3 years ago
Give me four reasons pluto is a cool planet / dwarf planet
AleksandrR [38]

Answer:

1. Just because it is small doesn't mean it needs to be excluded. If that were the case I would've been out of my friend group a while ago

2. Look at it it's fricking beautiful (see attachment)

3. It is just a great planet I don't think there needs to be any reason given I meannn you agree?

4. There's no other reason needed it's fricking gorgeous and amazing it needs no other reason. It needs a freaking

opening announcement to announce the arrival of the gorgeouness.

5 0
2 years ago
Read 2 more answers
Which planet has a tilted axis of rotation similar to that of earth, which means it has seasons?.
MrRa [10]

Answer:

Uranus

Explanation:

it's uranus....

3 0
2 years ago
Other questions:
  • At what angle North of East must the ship travel to reach its destination? Let East be 0◦ and North 90◦.
    12·1 answer
  • On takeoff, the combined action of the engines and wings of an airplane exerts a(n) 7934 N force on the plane, directed upward a
    11·1 answer
  • Charge is uniformly distributed throughout a spherical insulating volume of radius R=4.00 cm . The charge per unit volume is −6.
    14·1 answer
  • Can someone please help me out with this question thanks
    10·1 answer
  • A group of students wants to determine the internal resistance of a battery. They connect the battery to a variable resistor. Th
    15·1 answer
  • A 0.200 H inductor is connected in series with a 88.0 Ω resistor and an ac source. The voltage across the inductor is vL=−(12.0V
    6·1 answer
  • The half-life of Cs-137 is 30.2 years. If the initial mass of the sample is 1.00 kg, how much will remain after 151 years?
    13·2 answers
  • Name at least two other viruses that are at all time lows due to vaccines.
    5·1 answer
  • Can somone pls help me??!! i’m very stuck
    6·1 answer
  • The graph shows two runners participating in a race.
    6·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!