Attach a solid cylinder of mass M and radius R to a horizontal massless spring with spring constant k so that it can roll without slipping along a horizontal surface. If the system is released from rest at a position in which the spring is stretched by an amount x0 what is the period T of simple harmonic motion for the center of mass of the cylinder? Express your answer in terms of M and k (enter M for M, k for k and pi for π).
T=
T = 2*pi*sqrt((3*M)/(2*k))
To find the period T of simple harmonic motion for the center of mass of the cylinder attached to the spring, we need to use the formula for the period of a mass-spring system.
The period T is given by:
T = 2π * √(m_eff / k_eff)
where m_eff is the effective mass and k_eff is the effective spring constant of the system.
To solve for T, we need to find the expressions for m_eff and k_eff.
1. Effective Mass (m_eff):
The effective mass takes into account both the mass of the cylinder and the rotational inertia. For a solid cylinder rolling without slipping, the rotational inertia about its center of mass is given by:
I = 1/2 * M * R^2
where M is the mass of the cylinder and R is the radius. The effective mass is then calculated using the parallel-axis theorem, which relates the rotational inertia about the center of mass to the rotational inertia about a different axis at a distance h:
I_eff = I + m * h^2
Since the cylinder is rolling without slipping along a horizontal surface, the effective axis is given by I + m * R^2/2, where h = R/2. Therefore, we can substitute I_eff into the expression for m_eff:
m_eff = M + I / R^2
= M + (1/2 * M * R^2) / R^2
= M + 1/2 * M
= 3/2 * M
2. Effective Spring Constant (k_eff):
The effective spring constant takes into account both the linear spring constant and the rotational spring constant. For a rolling solid cylinder, the rotational spring constant is given by:
k_rot = I / R^2
We can substitute this rotational spring constant into the equation for k_eff:
k_eff = k + k_rot
= k + (1/2 * M * R^2) / R^2
= k + 1/2 * M
Now that we have expressions for m_eff and k_eff, we can substitute them into the equation for the period T:
T = 2π * √(m_eff / k_eff)
= 2π * √((3/2 * M) / (k + 1/2 * M))
= 2π * √(3 * M / (2 * k + M))
Therefore, the period T of simple harmonic motion for the center of mass of the cylinder is 2π * √(3 * M / (2 * k + M)).