Equivalent forces derivation problem.
EXAMPLE:
Derive the formula for F3 in terms of the experimentally measured quantities m1, m2, 1, and 2.
[Answer: F3=m1gcos1+m2gcos2.]
Make sure you understand how this formula was derived.
QUESTION:
If the mass of both weights is 150 gm, the first mass is located 20 degrees north of east, the second mass is located 20 degrees south of east, and the transducer sensitivity is 0.5 volts/Newton, how large a voltage do you expect to measure? Assume the transducer has been properly zeroed so that V = 0 when F3=0.
Without understanding the set-up, it is impossible to understand the questions.
To find the expected voltage output of the transducer, we need to first calculate the value of F3 using the given information.
From the given question, we know that:
mass of both weights (m1 and m2) = 150 gm
first mass location (θ1) = 20 degrees north of east
second mass location (θ2) = 20 degrees south of east
transducer sensitivity = 0.5 volts/Newton
To calculate F3, we can use the derived formula: F3 = m1gcosθ1 + m2gcosθ2
First, let's calculate the gravitational force acting on each mass. The acceleration due to gravity, g, is approximately 9.8 m/s^2.
For mass m1:
F1 = m1 * g = 0.15 kg * 9.8 m/s^2 = 1.47 N
For mass m2:
F2 = m2 * g = 0.15 kg * 9.8 m/s^2 = 1.47 N
Now, let's calculate the values of cosθ1 and cosθ2.
cosθ1 = cos(20 degrees north of east)
= cos(20 degrees)
≈ 0.9397
cosθ2 = cos(20 degrees south of east)
= cos(-20 degrees)
≈ 0.9397
Now substitute these values into the formula for F3:
F3 = m1gcosθ1 + m2gcosθ2
= 1.47 N * 0.9397 + 1.47 N * 0.9397
≈ 1.3839 N
Next, we can calculate the voltage output using the relationship given:
V = F3 * transducer sensitivity
V = 1.3839 N * 0.5 volts/Newton
≈ 0.69195 volts
Therefore, we expect to measure a voltage of approximately 0.69195 volts using the given values and formula.
To find the voltage expected to be measured, we need to first calculate the force F3 using the given information and then use the formula V = k * F3, where k is the transducer sensitivity.
Given information:
- Mass of both weights (m1 and m2): 150 gm
- Angle of weight 1 (1): 20 degrees north of east
- Angle of weight 2 (2): 20 degrees south of east
- Transducer sensitivity: 0.5 volts/Newton
To find the force F3, we can use the given formula F3 = m1gcos1 + m2gcos2, where m1 is the mass of weight 1, m2 is the mass of weight 2, g is the acceleration due to gravity, 1 is the angle of weight 1, and 2 is the angle of weight 2.
Plugging in the given values:
- m1 = m2 = 150 gm = 0.15 kg
- g = 9.8 m/s^2 (standard acceleration due to gravity, unless otherwise specified)
- 1 = 20 degrees north of east
- 2 = 20 degrees south of east
To find the force F3, we calculate:
F3 = (0.15 kg)(9.8 m/s^2) * cos(20 degrees) + (0.15 kg)(9.8 m/s^2) * cos(-20 degrees)
= (0.15 kg)(9.8 m/s^2) * cos(20 degrees) + (0.15 kg)(9.8 m/s^2) * cos(20 degrees) [using the property cos(-x) = cos(x)]
= 2 * (0.15 kg)(9.8 m/s^2) * cos(20 degrees)
Now, to find the voltage measured, we use the formula V = k * F3, where k is the transducer sensitivity (0.5 volts/Newton):
V = (0.5 volts/Newton) * (2 * (0.15 kg)(9.8 m/s^2) * cos(20 degrees))
= 0.5 * 2 * (0.15 kg)(9.8 m/s^2) * cos(20 degrees) volts
Now you can calculate the voltage by substituting the numerical values for the mass, acceleration due to gravity, and angle, and solving the equation.