A 10 kg block rests on a 30o inclined plane. The block is attached to a bucket by pulley system depicted below. The mass in the bucket is gradually increased by the addition of sand. At some point, the bucket will accumulate enough sand to set the block in motion. The coefficients of static and kinetic friction are 0.60 and 0.50 respectively.

Required:
a. Determine the mass of sand in [kg], including the bucket, needed to start the block moving.
b. Find the blocks acceleration, in [m/s^2] up the plane?

Answer:

a). M = 20.392 kg

b). am = 0.56 [tex]m/s^2[/tex] (block),  aM = 0.28 [tex]m/s^2[/tex] (bucket)

Explanation:

a). We got  N = mg cos θ,

                  f = [tex]$\mu_s N$[/tex]

                    = [tex]$\mu_s mg \cos \theta$[/tex]

If the block is ready to slide,

T = mg sin θ + f

T = mg sin θ + [tex]$\mu_s mg \cos \theta$[/tex]   .....(i)

2T = Mg ..........(ii)

Putting (ii) in (i), we get

[tex]$\frac{Mg}{2}=mg \sin \theta + \mu_s mg \sin \theta$[/tex]

[tex]$M=2(m \sin \theta + \mu_s mg \cos \theta)$[/tex]

[tex]$M=2 \times 10 \times (\sin 30^\circ+0.6 \cos 30^\circ)$[/tex]

M = 20.392 kg

b). [tex]$(h-x_m)+(h-x_M)+(h'+x_M)=l$[/tex]  .............(iii)

   Here, l = total string length

Differentiating equation (iii) double time w.r.t t, l, h and h' are constants, so

[tex]$-\ddot{x}-2\ddot x_M=0$[/tex]

[tex]$\ddot x_M=\frac{\ddot x_m}{2}$[/tex]

[tex]$a_M=\frac{a_m}{2}$[/tex]   .....................(iv)

We got,   N = mg cos  θ

                [tex]$f_K=\mu_K mg \cos \theta$[/tex]

∴ [tex]$T-(mg \sin \theta + f_K) = ma_m$[/tex]

  [tex]$T-(mg \sin \theta + \mu_K mg \cos \theta) = ma_m$[/tex]  ................(v)

Mg - 2T = M[tex]a_M[/tex]

[tex]$Mg-Ma_M=2T$[/tex]

[tex]$Mg-\frac{Ma_M}{2} = 2T$[/tex]    (from equation (iv))

[tex]$\frac{Mg}{2}-\frac{Ma_M}{4}=T$[/tex]   .....................(vi)

Putting (vi) in equation (v),

[tex]$\frac{Mg}{2}-\frac{Ma_M}{4}-mf \sin \theta-\mu_K mg \cos \theta = ma_m$[/tex]

[tex]$\frac{g\left[\frac{M}{2}-m \sin \theta-\mu_K m \cos \theta\right]}{(\frac{M}{4}+m)}=a_m$[/tex]

[tex]$\frac{9.8\left[\frac{20.392}{2}-10(\sin 30+0.5 \cos 30)\right]}{(\frac{20.392}{4}+10)}=a_m$[/tex]

[tex]$a_m= 0.56 \ m/s^2$[/tex]

Using equation (iv), we get,

[tex]a_M= 0.28 \ m/s^2[/tex]

If the distance between the center of two objects is quadrupled. The gravitational
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2. Our solar system is made up of the Sun, 8 planets, and other bodies such as
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Earth. The distance between planets is measured in astronomical units (AU). One AU is
equal to 149.6 million kilometers, the average distance between the Sun and Earth. Scale
models are useful for helping us understand the size of the solar system.
Mr. Wilson’s science class made a scale model of the solar system. They went out to the
school’s football field, and they used the chart shown below to mark out the scale distance
from the Sun to each planet

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For this project, you are expected to submit the following:
1. Your Student Guide with completed Student Worksheet
2. Your scale model of the solar system
Step 1: Prepare for the project.
a) Read through the guide before you begin so you know the expectations for this project.
b) If anything is not clear to you, be sure to ask your teacher.
Step 2: Conduct research on the actual sizes of the planets.
a) Do research to find the actual sizes of the Sun and the planets. This information is typically represented as diameter in kilometers (km). Recall that diameter is the length of the imaginary straight line from one side of a figure, such as a sphere, to the opposite side of the figure. This line passes through the center of the figure.
b) Record the actual diameters of the Sun and the planets in the first column of the table in the Student Worksheet.
c) Copy the link of the website you used into the space provided in the Student Worksheet.
Step 3: Determine the scaled sizes of the planets.
a) Go to a reliable website to find a solar system model calculator.
b) Decide how big you want the Sun in your model to be. For example, you could assign your Sun to be 300 mm. Input this figure in the calculator, and the calculator will determine the diameters of the eight planets for you. You want to make sure that the Sun is big enough so that the smallest planet will still be big enough to draw.
c) Record information from the calculator in the second column of the table in the Student Worksheet.
d) Copy the link of the website you used into the space provided in the Student Worksheet.
Step 4: Create a scale model of the solar system.
a) Draw and cut construction paper models of the Sun and the planets using the scaled measurements from the table.
b) Glue the models on the poster board. You can glue or tape poster boards together if necessary. Be sure to put the Sun in the center and to put the planets and a drawing of their orbits in order from nearest to farthest from the Sun.
Note: Remember that in this model, the diameter of the planets is scaled but the distance of the planets from the Sun is not. That means your model does not accurately represent the distances of the planets from the Sun so you need not worry about these measurements.
c) Label the Sun and the planets.
d) Put an attention-catching title above or below your model.
e) Write your name on the back of your poster board.
Step 5: Complete the Student Worksheet.
a) Make sure the table in the Student Worksheet is complete.
b) Answer the questions in the Student Worksheet.
c) Check to make sure you added the sources you used for this project in the Student Worksheet.
Step 6: Evaluate your project using this checklist.
If you can check each of the following boxes, you are ready to submit your project.
 Did you conduct research to find the actual size of the Sun and the planets? Did you record this information in the table in the Student Worksheet?
 Did you use a solar system model calculator to determine the scaled size of the Sun and planets? Did you record this information in the Student Worksheet?
 Did you add the links of the websites you used for this project to the Student Worksheet?
 Did you use the scaled sizes to create models of the Sun and the planets?
 Did you put your model together in a way that represents the solar system (Sun in the center and planets in order from nearest to farthest from the Sun)?
 Did you label each component of your model?
 Did you add an attention-catching title above or below your model?
 Did you write your name on the back of your poster board?
 Did you complete the Student Worksheet at the end of this guide?
Step 7: Revise and submit your project.
a) If you were unable to check off all the requirements on the checklist, go back and make sure that your project is complete. Save your project before submitting it.
b) Turn in your scale model of the solar system to your teacher. Be sure that your name is on it.
c) Submit your Student Guide through the virtual classroom.
d) Congratulations! You have completed your project.

Answer in your PE notebook
I have learned that
I have realized that
I will apply

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Time to travel
0.25 m
Final
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a = (v2 - v4)/(t2-tı)
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