A uniform rod of mass of 120g and a length of 130cm is suspended by a wire from a point 80cm from the rods left end . What mass must hang right end for the rod to be in equilibrium? What will be the mass of the wire?
Mass = 36g
Tension = 0.0156N
19.2 g
To find the mass required to hang on the right end of the rod for it to be in equilibrium, we need to consider the torque (rotational force) acting on the rod.
The torque depends on the mass, length, and distance from the fulcrum. In this case, the fulcrum is where the wire is attached to the rod.
Let's assume the mass required to hang on the right end of the rod is 'm' grams.
The torque due to the suspended mass on the right end is given by:
Torque_right = (m * 80) cm*g
The torque due to the rod's own weight (uniformly distributed) acts at the center of the rod, which is 65 cm from the right end. The torque due to the rod's weight is given by:
Torque_rod = (120 * 65) cm*g
For the rod to be in equilibrium, the torque on the left side due to the suspended mass must be equal to the torque on the right side due to the rod's weight.
Therefore, we can equate the two torques and solve for 'm':
(m * 80) = (120 * 65)
m = (120 * 65) / 80
m ≈ 97.5 grams
The mass required to hang on the right end of the rod for it to be in equilibrium is approximately 97.5 grams.
Now, let's calculate the mass of the wire. Since the wire is only acting as a support and not contributing to the rotational forces, its mass doesn't affect the equilibrium. Therefore, the mass of the wire is irrelevant in this context.
To determine the mass that must hang on the right end for the rod to be in equilibrium, we need to consider the rotational equilibrium of the system.
1. Start by calculating the torque exerted by the rod. Torque is given by the formula: Torque = Force × Distance × sin(θ), where θ is the angle between the force vector and the displacement vector.
2. The force acting on the rod is the weight of the rod, which is equal to the mass of the rod multiplied by the acceleration due to gravity. Weight (Force) = mass × acceleration due to gravity.
3. The distance at which the force is applied is the distance between the point of suspension and the center of mass of the rod. In this case, the distance is 80 cm.
4. The angle θ between the force and displacement vectors is 90 degrees because the force is acting vertically downwards while the displacement is horizontal.
5. With the equation Torque = Force × Distance × sin(θ), we can simplify it to: Torque = Force × Distance.
6. Now, let's consider the mass hanging on the right end. We'll denote it as M. The distance between this mass and the point of suspension is 130 cm minus 80 cm, which equals 50 cm.
7. The torque exerted by the hanging mass is equal to the weight of the mass multiplied by the distance. Torque = M × g × Distance, where g is the acceleration due to gravity.
8. In equilibrium, the net torque on the rod must be zero. So we can set the torques exerted by the rod and the hanging mass equal to each other: Torque exerted by the rod = Torque exerted by the hanging mass.
9. Now, we have an equation: Weight of the rod × Distance of the rod = Mass of the hanging mass × g × Distance of the hanging mass.
10. Substituting the known values, we have: (mass of the rod × g × Distance of the rod) = (mass of the hanging mass × g × Distance of the hanging mass).
11. Rearranging the equation, we find: mass of the hanging mass = (mass of the rod × Distance of the rod) / Distance of the hanging mass.
12. Plugging in the given values, we have: mass of the hanging mass = (120 g × 80 cm) / 50 cm.
Now, to calculate the mass of the wire, we need to subtract the mass of the rod and the hanging mass from the total mass of the system.
13. The total mass of the system is the sum of the mass of the rod and the hanging mass: Total mass = mass of the rod + mass of the hanging mass.
14. Plugging in the values, we have: Total mass = 120 g + mass of the hanging mass (obtained in the previous step).
By following these steps, you can find the mass of the hanging mass and the total mass of the system, including the wire.