A cylinder of radius R = 6.0 cm is on a rough horizontal surface. The coefficient of kinetic
friction between the cylinder and the surface is 0.30 and the rotational inertia for rotation
about the axis is given by MR2/2, where M is its mass. Initially it is not rotating but its
center of mass has a speed of 7.0 m/s. After 2.0 s the speed of its center of mass and its angular
velocity about its center of mass, respectively, are:
A. 1.1 m/s, 0
B. 1.1 m/s, 19 rad/s
C. 1.1 m/s, 98 rad/s
D. 1.1 m/s, 200 rad/s
E. 5.9 m/s, 98 rad/s
To solve this problem, we need to use the principles of rotational motion and friction. Let's break it down step by step:
Step 1: Calculate the initial linear momentum of the cylinder's center of mass.
The linear momentum is given by the equation p = mv, where p is momentum, m is mass, and v is velocity. In this case, the mass is not given. However, we can calculate it using the radius and the density of the cylinder.
Given: Radius (R) = 6.0 cm = 0.06 m
Density (ρ) is not provided.
Let's assume that the cylinder is made of a material with a typical density of 8,000 kg/m^3.
The volume (V) of the cylinder can be calculated using the formula V = πR^2h, where h is the height of the cylinder. Since the height is not given, we can assume it to be 2R.
V = π(0.06 m)^2(2R) = 0.072π m^3
The mass (m) can be calculated using the formula m = ρV.
Given: ρ = 8,000 kg/m^3
m = (8,000 kg/m^3)(0.072π m^3) ≈ 1,812 kg
Now, we can calculate the initial linear momentum.
Initial linear momentum (p_initial) = mv = (1,812 kg)(7.0 m/s) = 12,684 kg·m/s
Step 2: Calculate the net force acting on the cylinder.
The net force is given by Newton's second law, F = ma, where F is force, m is mass, and a is acceleration. In this case, we need to consider both translational and rotational motion.
The translational acceleration (a_translation) of the center of mass can be calculated using the equation a_translation = dv/dt.
Given: dv/dt = 7.0 m/s / 2.0 s = 3.5 m/s^2
The force (F_translation) causing this acceleration is given by F_translation = ma_translation.
F_translation = (1,812 kg)(3.5 m/s^2) = 6,342 N
The rotational force causing the acceleration is due to friction. The force of friction (F_friction) can be calculated using the equation F_friction = μN, where μ is the coefficient of kinetic friction and N is the normal force.
The normal force (N) is equal to the weight of the cylinder, which can be calculated using the equation N = mg, where g is the acceleration due to gravity.
Given: μ = 0.30, g = 9.8 m/s^2
N = mg = (1,812 kg)(9.8 m/s^2) = 17,805.6 N
F_friction = μN = (0.30)(17,805.6 N) = 5,341.7 N
Note: The force of friction acts in the direction opposite to the motion.
The net force (F_net) is the vector sum of F_friction and F_translation.
F_net = F_translation - F_friction = 6,342 N - 5,341.7 N = 1,000.3 N
Step 3: Calculate the angular acceleration.
The angular acceleration (α) can be calculated using the equation α = τ/I, where τ is the net torque and I is the moment of inertia.
The net torque acting on the system is equal to the product of the force of friction (F_friction) and the radius of the cylinder (R).
τ = F_friction × R = (5,341.7 N)(0.06 m) = 320.5 N·m
The moment of inertia (I) for a cylinder rotating about its axis is given by I = MR^2/2, where M is the mass.
I = (1,812 kg)(0.06 m)^2/2 = 3.091 kg·m^2
Now we can calculate the angular acceleration.
α = τ/I = 320.5 N·m / 3.091 kg·m^2 = 103.78 rad/s^2
Step 4: Calculate the final speed of the center of mass.
Using the equation v_final = v_initial + at, we can determine the final velocity of the center of mass after 2.0 seconds.
Given: t = 2.0 s, dv/dt = 3.5 m/s^2 (from earlier calculations)
v_final = v_initial + dv/dt × t = 7.0 m/s + (3.5 m/s^2)(2.0 s) = 7.0 m/s + 7.0 m/s = 14.0 m/s
Step 5: Calculate the final angular velocity.
The final angular velocity (ω_final) can be calculated using the equation ω_final = ω_initial + αt, where ω_initial is the initial angular velocity and t is the time.
Given: ω_initial = 0 rad/s (initially not rotating), t = 2.0 s, α = 103.78 rad/s^2 (from earlier calculations)
ω_final = ω_initial + αt = 0 rad/s + (103.78 rad/s^2)(2.0 s) = 0 rad/s + 207.56 rad/s = 207.56 rad/s
Step 6: Finalize the answer.
The final speed of the center of mass is 14.0 m/s, and the final angular velocity is 207.56 rad/s.
Therefore, the correct answer is E. 14.0 m/s, 207.56 rad/s.