A particle of mass 0.350 kg is attached to the 100-cm mark of a meter stick of mass 0.175 kg. The meter stick rotates on a frictionless, horizontal table with an angular speed of 6.00 rad/s.
(a) Calculate the angular momentum of the system when the stick is pivoted about an axis perpendicular to the table through the 50.0-cm mark.
kg · m2/s
(b) Calculate the angular momentum of the system when the stick is pivoted about an axis perpendicular to the table through the 0-cm mark.
kg · m2/s
To find the angular momentum of the system, we need to know the formula for angular momentum and how it is calculated.
Angular momentum (L) is given by the product of the moment of inertia (I) and the angular velocity (ω) of an object:
L = I * ω
where L is measured in kg · m^2/s, I is the moment of inertia, and ω is the angular velocity in radians per second.
(a) To find the angular momentum when the stick is pivoted about an axis perpendicular to the table through the 50.0-cm mark, we need to calculate the moment of inertia of the system.
The moment of inertia of a system is the sum of the individual moments of inertia of each object. For the meter stick, the moment of inertia (I_stick) is given by:
I_stick = (1/12) * M_stick * L_stick^2
where M_stick is the mass of the meter stick and L_stick is its length. Given that M_stick = 0.175 kg and L_stick = 100 cm = 1 m, we can calculate I_stick as:
I_stick = (1/12) * 0.175 kg * (1 m)^2
Once we have the moment of inertia of the stick, we can find the moment of inertia of the particle (I_particle) by using the parallel axis theorem:
I_particle = I_particle_cm + M_particle * d^2
where I_particle_cm is the moment of inertia of the particle about its center of mass, M_particle is the mass of the particle, and d is the distance of the particle from the axis of rotation. Given that M_particle = 0.350 kg and d = 50 cm = 0.5 m, we can calculate I_particle_cm as:
I_particle_cm = M_particle * r^2
where r is the distance of the particle from its center of mass. Given that r = L_stick/2 = 0.5 m, we can calculate I_particle_cm as:
I_particle_cm = 0.350 kg * (0.5 m)^2
Now we can calculate the total moment of inertia of the system (I_system) by summing up the individual moments of inertia:
I_system = I_stick + I_particle
Finally, we can calculate the angular momentum (L) of the system by multiplying the moment of inertia (I_system) by the angular velocity (ω):
L = I_system * ω
(b) To find the angular momentum when the stick is pivoted about an axis perpendicular to the table through the 0-cm mark, the procedure is the same as in part (a), but we need to recalculate the moment of inertia of the particle and the total moment of inertia of the system considering the new pivot point. The rest of the calculation remains the same.