A uniform metal rod of length 86cm and mass 3.2kg is suspended by two vertical springs balances c and d.Balance c is 20cm from one end while balance d is 30cm from the other end.Find the reading on each balance
To find the readings on each balance, we need to analyze the equilibrium of forces acting on the metal rod.
Let's assume that the left end of the rod is end A, and the right end is end B. Balance C is located 20 cm from end A, and balance D is located 30 cm from end B.
First, let's find the center of mass of the rod. Since the rod is uniform, the center of mass will be located at the midpoint of the rod, which is 43 cm from either end (86 cm / 2).
To maintain the equilibrium, the torques acting on the rod about the left end must be equal to the torques acting on the rod about the right end.
Torque = Force × Distance
Let's assume the reading on balance C is Fc, and the reading on balance D is Fd. The torques can be calculated as follows:
Torque about end A = Fc × 20 cm
Torque about end B = Fd × 30 cm
Since the rod is in equilibrium, these torques must be equal:
Fc × 20 cm = Fd × 30 cm
Now, let's consider the weight of the rod. The weight acts at the center of mass of the rod, which is 43 cm from either end. The weight can be calculated using the formula:
Weight = mass × gravity
In this case, the mass of the rod is given as 3.2 kg, and gravity is approximately 9.8 m/s^2.
Weight of the rod = 3.2 kg × 9.8 m/s^2
Once we have the weight of the rod, we can distribute it between the two balances. Since the rod is in equilibrium, the sum of the forces on the balances must be equal to the weight of the rod.
Fc + Fd = Weight of the rod
Now, we have two equations:
Fc × 20 cm = Fd × 30 cm
Fc + Fd = Weight of the rod
We can solve these equations simultaneously to find the readings on each balance.
1. Substitute the weight of the rod into the second equation to get:
Fc + Fd = 3.2 kg × 9.8 m/s^2
2. Rearrange the first equation Fc × 20 cm = Fd × 30 cm to get:
Fc = (Fd × 30 cm) / 20 cm
3. Substitute this expression for Fc into the second equation:
(Fd × 30 cm) / 20 cm + Fd = 3.2 kg × 9.8 m/s^2
4. Convert the centimeters to meters:
(Fd × 0.3 m) / 0.2 m + Fd = 3.2 kg × 9.8 m/s^2
5. Simplify the equation:
1.5 Fd + Fd = 3.2 kg × 9.8 m/s^2
6. Combine like terms:
2.5 Fd = 3.2 kg × 9.8 m/s^2
7. Solve for Fd:
Fd = (3.2 kg × 9.8 m/s^2) / 2.5
Now, substitute the value of Fd back into the first equation to find Fc:
Fc × 20 cm = Fd × 30 cm
Fc = (Fd × 30 cm) / 20 cm
Finally, we can substitute the values of Fc and Fd into the original equations to find the readings on balance C and balance D. Remember to convert the forces from Newtons to grams to match the units of the original problem.
Reading on balance C = Fc × 1000 g
Reading on balance D = Fd × 1000 g
By following these steps, you can find the readings on each balance of the suspended metal rod.