Four masses are arranged as shown. They are connected by rigid, massless rods of lengths 0.50 m and d = 0.73 m. What torque must be applied to cause an angular acceleration of 0.63 rad/s2 about the axis shown?
4 kg
3 kg
5 kg
2 kg
I=(0.73/2)²(4+3+5+2) = 1.87 kg•m²
M=Iε=1.87•0.63 =1.175 kg•m
To find the torque required to cause the given angular acceleration, we need to calculate the moment of inertia for each mass and then apply the rotational analogue of Newton's second law, which states that the torque τ applied to an object is equal to the moment of inertia I multiplied by the angular acceleration α.
First, we need to calculate the moment of inertia for each mass. The moment of inertia for a point mass rotating about an axis perpendicular to its motion and passing through its center of mass is given by the formula:
I = m * r^2
Where I is the moment of inertia, m is the mass, and r is the perpendicular distance between the axis of rotation and the mass.
For the masses given:
Mass 1 (4 kg): moment of inertia I1 = m1 * r1^2 = 4 kg * (0.50 m)^2 = 4 kg * 0.25 m^2 = 1 kg·m^2
Mass 2 (3 kg): moment of inertia I2 = m2 * r2^2 = 3 kg * (0.50 m + 0.73 m)^2 = 3 kg * 1.23 m^2 = 3.69 kg·m^2
Mass 3 (5 kg): moment of inertia I3 = m3 * r3^2 = 5 kg * (0.50 m + 0.73 m)^2 = 5 kg * 1.23 m^2 = 6.15 kg·m^2
Mass 4 (2 kg): moment of inertia I4 = m4 * r4^2 = 2 kg * (0.50 m + 0.73 m)^2 = 2 kg * 1.23 m^2 = 2.46 kg·m^2
The total moment of inertia for the system is the sum of the individual moment of inertia values:
I_total = I1 + I2 + I3 + I4 = 1 kg·m^2 + 3.69 kg·m^2 + 6.15 kg·m^2 + 2.46 kg·m^2 = 13.3 kg·m^2
Now we can calculate the torque required using the formula τ = I_total * α:
τ = 13.3 kg·m^2 * 0.63 rad/s^2 = 8.379 N·m
Therefore, the torque that must be applied to cause an angular acceleration of 0.63 rad/s^2 about the given axis is approximately 8.379 N·m.
To determine the torque required to cause an angular acceleration, we need to calculate the moment of inertia of the system and then multiply it by the angular acceleration.
Step 1: Calculate the moment of inertia of each mass.
The moment of inertia of a point mass is given by the formula:
I = m * r^2
where I is the moment of inertia, m is the mass, and r is the distance from the axis of rotation.
For the 4 kg mass:
I1 = (4 kg) * (0.50 m)^2 = 4 kg * 0.25 m^2 = 1 kg*m^2
For the 3 kg mass:
I2 = (3 kg) * (0.73 m)^2 = 3 kg * 0.5329 m^2 = 1.5987 kg*m^2
For the 5 kg mass:
I3 = (5 kg) * (0.50 m + 0.73 m)^2 = 5 kg * 1.23 m^2 = 6.15 kg*m^2
For the 2 kg mass:
I4 = (2 kg) * (0.50 m + 0.73 m)^2 = 2 kg * 1.23 m^2 = 2.46 kg*m^2
Step 2: Calculate the total moment of inertia.
Since the masses are arranged in a particular configuration, we need to add the individual moments of inertia to obtain the total moment of inertia of the system.
I_total = I1 + I2 + I3 + I4
I_total = 1 kg*m^2 + 1.5987 kg*m^2 + 6.15 kg*m^2 + 2.46 kg*m^2
I_total = 11.1987 kg*m^2
Step 3: Calculate the torque.
The torque can be calculated using the formula:
τ = I * α
where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
τ = (11.1987 kg*m^2) * (0.63 rad/s^2)
τ = 7.0508 N*m
Therefore, the torque that must be applied to cause an angular acceleration of 0.63 rad/s^2 about the given axis is 7.0508 N*m.