A uniform rod of length 80cm and a mass of 3.2 kg is supported horizontally by to vertical spring balances C and D. Balance C is 20 cm from one end and balance D is 30 cm from the other end. Find the reading on each balance.
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To find the reading on each balance, we can use the principle of torque equilibrium.
Let's consider balance C first, which is 20 cm from one end of the rod. The weight of the rod (mg) acts at the center of gravity of the rod, which is located at 40 cm from either end of the rod (80 cm / 2 = 40 cm).
Now, let's calculate the torques acting on balance C:
- The weight of the rod creates a clockwise torque around balance C. The torque is given by: TC = (Weight of the rod) * (Distance between balance C and center of gravity).
The weight of the rod is given by the formula: Weight = mass * gravity
The distance between balance C and the center of gravity is 20 cm.
Now, let's calculate the torque TC:
Weight = mass * gravity
Weight = 3.2 kg * 9.8 m/s^2
Weight = 31.36 N
TC = Weight * Distance
TC = 31.36 N * 0.2 m
TC = 6.27 Nm
Therefore, the torque acting on balance C is 6.27 Nm.
Now, let's consider balance D, which is 30 cm from the other end of the rod. The distance between balance D and the center of gravity is 50 cm.
Similarly, the torque TD acting on balance D is given by:
Weight = mass * gravity
Weight = 3.2 kg * 9.8 m/s^2
Weight = 31.36 N
TD = Weight * Distance
TD = 31.36 N * 0.3 m
TD = 9.41 Nm
Therefore, the torque acting on balance D is 9.41 Nm.
To find the reading on each balance, we need to consider that the sum of the torques on both sides of the rod should be zero to maintain equilibrium:
TC + TD = 0
Substituting the values we've found:
6.27 Nm + 9.41 Nm = 0
Therefore, the reading on each balance is:
Balance C: -6.27 Nm
Balance D: -9.41 Nm
The negative sign indicates that the reading on balance D is in the opposite direction compared to balance C and the weight of the rod.
To find the reading on each balance, we need to consider the forces acting on the rod and the equilibrium conditions.
Let's denote the length of the rod as L = 80 cm = 0.8 meters and its mass as m = 3.2 kg.
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. So, the center of mass is at a distance of L/2 = 0.4 meters from either end of the rod.
Now, let's set up the equilibrium conditions based on torques acting on the rod. The net torque acting on the rod must be zero for it to be in rotational equilibrium.
The torque due to the force exerted by spring balance C (TC) can be calculated as follows:
TC = force on balance C × perpendicular distance between the force and the axis of rotation
The force on balance C can be denoted as FC, and the perpendicular distance between balance C and the center of mass is (L/2 - 20 cm) = 0.4 m - 0.2 m = 0.2 m.
Similarly, the torque due to the force exerted by spring balance D (TD) can be calculated as:
TD = force on balance D × perpendicular distance between the force and the axis of rotation
The force on balance D can be denoted as FD, and the perpendicular distance between balance D and the center of mass is (L/2 - 30 cm) = 0.4 m - 0.3 m = 0.1 m.
Since the rod is in equilibrium, the net torque acting on it is zero:
TC + TD = 0
Substituting the torque expressions:
FC × 0.2 m + FD × 0.1 m = 0
Now, let's consider the force balance along the rod. The sum of all the forces acting on the rod must be zero for it to be in translational equilibrium.
The forces acting on the rod are the forces exerted by spring balances C and D (FC and FD) and the weight of the rod (mg), acting at its center of mass.
Since the rod is in equilibrium, the net force acting on it is zero:
FC + FD - mg = 0
Substituting the values, we have:
FC + FD - 3.2 kg × 9.8 m/s^2 = 0
We have two equations:
1) FC × 0.2 m + FD × 0.1 m = 0 (Equation 1)
2) FC + FD - 3.2 kg × 9.8 m/s^2 = 0 (Equation 2)
Now, we'll solve the equations to find the readings on each balance.
From Equation 1, we have FC × 0.2 m = - FD × 0.1 m.
Dividing both sides by 0.2 m, we get:
FC = - 0.5 × FD
Substituting this into Equation 2, we have:
-0.5 × FD + FD - 3.2 kg × 9.8 m/s^2 = 0
Combining like terms:
0.5 × FD = 3.2 kg × 9.8 m/s^2
Dividing both sides by 0.5, we get:
FD = (3.2 kg × 9.8 m/s^2) / 0.5
Calculating the value:
FD = 64 N
Now, we can substitute this value into Equation 1 to find FC:
FC × 0.2 m = - FD × 0.1 m
FC × 0.2 m = - 64 N × 0.1 m
Dividing both sides by 0.2 m:
FC = - 64 N × 0.1 m / 0.2 m
Calculating the value:
FC = - 32 N
Therefore, the reading on balance C is -32 N and the reading on balance D is 64 N.
Note: The negative sign for balance C indicates that it is reading in the opposite direction to balance D, as expected since the rod is supported at different distances from each end.