A 240-g
mass hangs from a string that is wrapped around a pulley, as shown in the figure. The pulley is suspended in such a way that it can rotate freely. When the mass is released, it accelerates toward the floor as the string unwinds. Model the pulley as a uniform solid cylinder of mass 1.00 kg
and radius 7.00 cm.
Assume that the thread has negligible mass and does not slip or stretch as it unwinds.
To model the system and calculate the acceleration of the mass as it falls, we need to consider the forces acting on it and the pulley.
First, the gravitational force acting on the mass can be calculated using the formula:
F_gravity = m * g
where m is the mass of the object (240g or 0.240 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Next, we need to consider the rotational dynamics of the pulley. As the mass falls and the string unwinds, the pulley will rotate. The net torque on the pulley is given by the difference between the torque of the object hanging on the string and the torque due to the mass of the pulley itself.
The torque caused by the hanging mass is given by:
τ_hanging_mass = r * F_gravity
where r is the radius of the pulley (7.00 cm or 0.07 m).
The torque due to the pulley's mass can be calculated using the formula for the moment of inertia:
τ_pulley_mass = I * α
where I is the moment of inertia of a solid cylinder given by:
I = (1/2) * m_pulley * r^2
and α is the angular acceleration.
Now, since the string is assumed to have no mass and not to slip or stretch, the linear acceleration of the mass is equal to the tangential acceleration of the pulley's edge. This can be expressed as:
a_tangential = α * r
Finally, the linear acceleration of the mass can be calculated using the following equation of motion:
a_mass = a_tangential
Now, combining all the equations, we can solve for the linear acceleration of the mass.