A body mass Xg suspended at the 15cm mark of a ruler balanced it at the 30cm mark on a pivot .If the ruler has a mass of 36g and it's centre of gravity is at 55cm mark, calculate the mass X and the balance point of the ruler from zero and when the body is moved to the 25cm mark
Solve the question in detail please
Solve the question in detail please
Well, it seems we have a balancing act here! Let's crunch some numbers, shall we?
To find the mass X, we need to first determine the moment on each side of the pivot. The total moment on one side should be equal to the total moment on the other side for perfect balance.
On one side, we have the ruler with a mass of 36g and its center of gravity at the 55cm mark. So the moment on that side is (36g) * (55cm).
On the other side, we have the body mass Xg suspended at the 15cm mark. So the moment on that side is (Xg) * (15cm).
Since the ruler is balanced at the 30cm mark, the moments on both sides are equal. Therefore:
(36g) * (55cm) = (Xg) * (15cm)
Now we can solve for X:
(36g * 55cm) / 15cm = Xg
X = (36 * 55) / 15
X ≈ 132g
So the mass X is approximately 132g.
Now let's find the new balance point when the body is moved to the 25cm mark.
Again, the sum of the moments on both sides should be equal. On one side, we have the ruler with mass 36g and its center of gravity at the 55cm mark. The moment on this side is (36g) * (55cm).
On the other side, now with the body at the 25cm mark, the moment is (Xg) * (25cm).
Setting the moments equal to each other:
(36g) * (55cm) = (Xg) * (25cm)
Now we can solve for the new balance point:
(36g * 55cm) / (Xg) = 25cm
25cm * (Xg) = (36g * 55cm)
X ≈ (36 * 55) / 25
X ≈ 79g (approximately)
So the new balance point from zero when the body is moved to the 25cm mark is approximately 79cm.
I hope I haven't turned your brain into a balancing act with all these numbers! If you have any more questions, feel free to ask!
To calculate the mass of the body, we can make use of the principle of moments, which states that the sum of the clockwise moments is equal to the sum of the anticlockwise moments when an object is in equilibrium.
In this case, the ruler is in equilibrium on the pivot. We can calculate the counterclockwise moments (due to the body X) and the clockwise moments (due to the ruler itself) and equate them to find the unknowns.
Let's use the following variables:
- Mass of the body, X = mX
- Distance of the body from the pivot = dX = 15 cm
- Mass of the ruler = mR = 36 g
- Distance of the ruler's center of gravity from the pivot = dR = 55 cm
- Distance of the balance point of the ruler from the pivot = dB
To find the mass, X, we can start by calculating the moments.
The moment due to the body X is calculated as: Moment_X = mX * dX
The moment due to the ruler is calculated as: Moment_R = mR * dR
Since the ruler is balanced, the sum of the clockwise moments must be equal to the sum of the anticlockwise moments:
Moment_R + Moment_X = 0
Substituting the values we have:
mR * dR + mX * dX = 0
36g * 55cm + mX * 15cm = 0
1980g + 15mX = 0
15mX = -1980g
mX = -1980g / 15
mX ≈ -132g
The negative sign indicates that the mass is in the opposite direction of the ruler's moment.
Now, let's calculate the balance point of the ruler when the body is moved to the 25 cm mark.
To find the new balance point, we need to consider the moments (torques) due to the body X and the ruler.
Moment_X = mX * dX_new
Moment_R = mR * dR_new
Since the ruler is still balanced, the sum of the clockwise moments must be equal to the sum of the anticlockwise moments:
Moment_R + Moment_X = 0
mR * dR_new + mX * dX_new = 0
Substituting the values we have:
36g * dB + (-132g) * 25cm = 0
36g * dB + (-3300gcm) = 0
36g * dB = 3300gcm
dB = 3300gcm / 36g
dB ≈ 91.67cm
Therefore, when the body is moved to the 25cm mark, the balance point of the ruler from zero is approximately 91.67 cm.