Question: User: The Builder
You are the builder of a balanced force system. Apply a mass ($55g$) to the M1 mass hanger at an angle ($\theta$) (determined by you).
Calculate and record the magnitude of $F$ (based on $F = mg$). Use the Force Wheel to measure the angle ($\theta$) of $F$.
Using Sine and Cosine functions, calculate the x-component and y-components of the Force that youe created.
Force x-component $0.40 N$ Force y-component $0.36 N$
Now attempt to create balance to the Force that youe created by increasing the tension in the spring as well as adding mass to the M2 mass hanger. Change only these two factors until the force disc is balanced in the center of the wheel.
What is the measurement reading from the Spring Scale?
Fx = $0.4 N$
What is the Force required on M2 in order to balance the force wheel?
Fy = $0.36 N$
Are these measured values similar to the calculated values?
Be sure youe taken photos of your setup.
Note: These x- and y-components are actually the x- and y-components of the equilibrant, $F_E$, of the force $F$, rather than the components of $F$ itself (acting in equal and opposite directions).
Graph/shape: instructional bullet-point text with two highlighted note lines; position_hint=top-left
calculate the math and show me step by step how to do it
1. **State the problem:**
We have a mass $m = 55g = 0.055 kg$ applied to the M1 mass hanger at an angle $\theta$. We want to calculate the magnitude of the force $F$ due to gravity, then find the x- and y-components of the force using sine and cosine functions. Finally, we want to balance this force by adjusting the tension and mass on M2.
2. **Calculate the magnitude of the force $F$:**
The force due to gravity is given by:
$$F = mg$$
where $g = 9.8 \frac{m}{s^2}$ is the acceleration due to gravity.
Calculate:
$$F = 0.055 \times 9.8 = 0.539 N$$
3. **Given components of the equilibrant force $F_E$:**
- $F_{x} = 0.40 N$
- $F_{y} = 0.36 N$
These are the components of the equilibrant force, which balances $F$.
4. **Calculate the angle $\theta$ of the force $F$ using components:**
Use the tangent function:
$$\tan(\theta) = \frac{F_y}{F_x} = \frac{0.36}{0.40} = 0.9$$
Calculate $\theta$:
$$\theta = \arctan(0.9) \approx 42^\circ$$
5. **Calculate the magnitude of the equilibrant force $F_E$ from components:**
$$F_E = \sqrt{F_x^2 + F_y^2} = \sqrt{0.40^2 + 0.36^2} = \sqrt{0.16 + 0.1296} = \sqrt{0.2896} \approx 0.538 N$$
This matches the magnitude of $F$ calculated in step 2, confirming balance.
6. **Check if measured values are similar to calculated values:**
- Calculated $F = 0.539 N$
- Calculated $F_E = 0.538 N$
- Given $F_x = 0.40 N$, $F_y = 0.36 N$
These values are very close, indicating the system is balanced.
7. **Summary:**
- The force magnitude $F$ from the mass is approximately $0.54 N$.
- The equilibrant force components are $F_x = 0.40 N$ and $F_y = 0.36 N$.
- The angle of the force is approximately $42^\circ$.
- The measured and calculated values agree, confirming the balance of the force system.
This completes the calculation and verification of the balanced force system.