Please refer to the picture with the formulas needed to solve this problem. Please only use the formulas, constants, and units from the picture given.
To simulate various processes on the surface on Mars, a labratory uses an Atwood machine to simulate Martian gravitational acceleration. An Atwood machine is a setup involving two objects connected by a string that runs over a pulley. For this problem, assume that the string and pulley are ideally frictionless, massless, and inextensible.
One object is a bucket holding the subject of the labratory's experiment, which in total has a mass mb. The other object is a counterweight, whose mass Mc must be properly selected to ensure that the experiement's downward acceleration matches that of Martian gravity, gM.
Part 1) The experimenter want the bucket to be accelerated as if it were being dropped near the surface of Mars. Which mass must be bigger, Mb (the bucket) or Mc (the counterweight)? How does the acceleration of the bucket compare to the acceleration of the counterweight?
Answer: The bucket have acceleration which is gM that is acceleration on mars.
Hence,
a = gM
Since the bucket needs to go down as if it were free falling on mars,
Hence, the mass of the bucket Mb needs to be bigger than counterweight Mc.
Part 2) Draw and label a free body diagram for both the experiment bucket and the counterweight.
Answer: On the bucket forces are gravitation force of earth downwards = Mb g and tension T upwards. On counterweight, its weight is acting downwards = Mc g. The tension is acting upwards.
Part 3) Write a Newton's 2nd Law equation describing the net force on the bucket.
Answer: The acceleration of bucket is a=gM
The net force on bucket is:
F = Mb g - T
Hence the second law of motion is as:
Mbg - T = MbgM
Part 4) Write a Newton's 2nd law equation describing the net force on the counterweight.
Part 5) If mb = 130 kg, determine the required counterweight mass Mc to achieve an acceleration equivalent to Martian gravity, gM = 3.7 m/s^2.
Please answer part 4 and part 5. Please answer with the correct formula from this formula sheet. Thank you
![## Educational Reference Page
### Formulas
1. **Average Acceleration**
\[
\vec{a}_{\text{avg}} = \frac{\Delta \vec{V}}{\Delta t} \rightarrow \vec{V}_f = \vec{V}_i + \vec{a}_{\text{avg}} \Delta t
\]
2. **Average Velocity**
\[
\bar{V}_{\text{avg}} = \frac{1}{2} (\vec{V}_i + \vec{V}_f)
\]
3. **Average Velocity with Displacement**
\[
\bar{V}_{\text{avg}} = \frac{\Delta \vec{x}}{\Delta t} \rightarrow \vec{x}_f = \vec{x}_i + \vec{V}_i \Delta t + \frac{1}{2} \vec{a} \Delta t^2
\]
4. **Kinematic Equation**
\[
V_f^2 = V_i^2 + 2a\Delta x
\]
5. **Force**
\[
F = m \vec{a}
\]
6. **Force of Friction**
\[
F_f = \mu_k F_N
\]
7. **Radial Acceleration**
\[
a_R = \frac{V^2}{r}
\]
8. **Newton's Third Law**
\[
\vec{F}_{AB} = -\vec{F}_{BA}
\]
9. **Frictional Force Condition**
\[
F_p \leq \mu_s F_N
\]
10. **Gravitational Force**
\[
F_G = G \frac{m_1 m_2}{r^2}
\]
11. **Frequency**
\[
f = \frac{1}{T}
\]
### Constants
1. Speed of Light: \( c \approx 3 \times 10^8 \, \text{m/s} \)
2. Gravitational Acceleration: \( g \approx 9.8 \, \text{m/s}^2 \)
3. Gravitational Constant: \( G \approx 6.7 \times 10^{-11} \,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F043a0ede-8e66-4ff6-855d-c3c5ac86ef03%2Fb1224d01-14dd-4e1e-8de5-88823c0502c5%2F3hrhqgt_processed.jpeg&w=3840&q=75)
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