Three masses of 4.0 kg, 5.0 kg and 7.0 kg are located on a line at positions +4.0, +8.0, and -5.0 in meters from the origin, respectively. Where is the center of mass of these three masses?
With respect to the origin as x = 0, I calculated the center of mass to be:
(4.0 kg)*(4.0 m)+(5.0 kg)*(8.0 m)+(7.0 kg)*(-5.0 m), all divided by the total mass of 16 kg. This gives an answer of 1.3125 m to the positive side of the origin.
However the answer in the book says the centre of mass is located at 1.0 m. I don't know where I went wrong.
(4 * 4) + (8 * 5) > 7 * 5
... so the CM is > 0
[4*(4-x)]+[5*(8-x)]-[7*(5+x)] = 0
Well, it seems like there's some confusion going on here. But don't worry, I'm here to help you figure it out with a touch of humor!
It looks like you've done the math correctly. But who am I to argue with the book? Maybe it has a different sense of humor than I do!
Now, let's solve this mystery together. To find the center of mass, we need to take into account both the position and the mass of each object. So, let's review the given information:
- Mass of the first object (m1): 4.0 kg
- Position of the first object (x1): +4.0 m
- Mass of the second object (m2): 5.0 kg
- Position of the second object (x2): +8.0 m
- Mass of the third object (m3): 7.0 kg
- Position of the third object (x3): -5.0 m
Now, to find the center of mass, we can use the formula:
Center of mass (x_cm) = (m1 * x1 + m2 * x2 + m3 * x3) / (m1 + m2 + m3)
Plugging in the values we have:
x_cm = (4.0 kg * 4.0 m + 5.0 kg * 8.0 m + 7.0 kg * -5.0 m) / (4.0 kg + 5.0 kg + 7.0 kg)
Now, let's do the math:
x_cm = (16.0 m² + 40.0 m² - 35.0 m²) / 16.0 kg
= 21.0 m² / 16.0 kg
≈ 1.313 m
So, based on my hilarious calculations, the center of mass is indeed approximately 1.313 m from the origin. Now, let's give the book the benefit of the doubt and assume it just rounded down to 1.0 m. Books can be quite... "exact" sometimes!
I hope this clarifies things for you. Remember, physics is a great balancing act, just like a clown on a unicycle juggling rubber chickens. Keep up the good work!
To find the center of mass, you need to take into account both the masses and their respective positions. It appears that you have correctly calculated the numerator of the equation, which represents the sum of the individual masses multiplied by their respective positions.
However, you made a mistake when dividing by the total mass (16 kg). The center of mass is found by dividing the sum of the individual masses multiplied by their positions by the total mass. The correct calculation should be:
[(4.0 kg * 4.0 m) + (5.0 kg * 8.0 m) + (7.0 kg * -5.0 m)] / (4.0 kg + 5.0 kg + 7.0 kg)
Now, we can simplify this equation and calculate the final result:
[(16.0 kg m) + (40.0 kg m) + (-35.0 kg m)] / 16.0 kg + 5.0 kg + 7.0 kg
[21.0 kg m] / 16.0 kg + 12.0 kg
1.3125 m / 16.0 kg + 12.0 kg
0.08203125 m / kg
Hence, the correct center of mass is approximately 0.082 m, not 1.3125 m as you calculated. It seems that the answer in the book is incorrect, and your calculation is indeed accurate.
To find the center of mass of a system of masses, you need to take into account both the masses and their respective positions.
The formula for calculating the center of mass is:
x_cm = (m1*x1 + m2*x2 + m3*x3) / (m1 + m2 + m3)
Where:
x_cm is the position of the center of mass
m1, m2, m3 are the masses of the three objects
x1, x2, x3 are the positions of the three objects
Let's calculate the center of mass using the given values:
m1 = 4.0 kg, x1 = 4.0 m
m2 = 5.0 kg, x2 = 8.0 m
m3 = 7.0 kg, x3 = -5.0 m
x_cm = (4.0 kg * 4.0 m + 5.0 kg * 8.0 m + 7.0 kg * (-5.0 m)) / (4.0 kg + 5.0 kg + 7.0 kg)
= (16.0 kg * m + 40.0 kg * m - 35.0 kg * m) / 16.0 kg
= (21.0 kg * m) / 16.0 kg
≈ 1.3125 m
Your calculation is correct, and the center of mass is indeed located at approximately 1.3125 m to the positive side of the origin.
It's possible that the answer in the book is rounded to 1.0 m, but the more precise value is 1.3125 m based on the given information.