A yoyo of mass m=1 kg and moment of inertia ICM=0.045 kg m2 consists of two solid disks of radius R=0.3 m, connected by a central spindle of radius r=0.225 m and negligible mass. A light string is coiled around the central spindle. The yoyo is placed upright on a flat rough surface and the string is pulled with a horizontal force F=20 N, and the yoyo rolls without slipping.
(a) What is the x-component of the acceleration of the center of mass of the yoyo? (in m/s2)
a=
(b) What is the x-component of the friction force? (in N)
f=
To solve this problem, we'll need to use the equations of rotational motion and Newton's second law.
(a) The x-component of the acceleration of the center of mass of the yoyo can be found using the equation:
a_cm = (F - f) / m
where F is the applied force, f is the friction force, and m is the mass of the yoyo.
(b) The x-component of the friction force can be found using the equation for the torque:
τ = ICM * α
where τ is the torque, ICM is the moment of inertia of the yoyo, and α is the angular acceleration.
First, let's calculate the angular acceleration:
α = a_cm / r
where r is the radius of the central spindle.
Now, we can calculate the torque:
τ = ICM * α
Next, we'll use the relationship between the torque and the friction force:
τ = r * f
Finally, we can solve for the friction force:
f = τ / r
Let's substitute the given values into the equations.
Given:
m = 1 kg
ICM = 0.045 kg m^2
F = 20 N
R = 0.3 m
r = 0.225 m
(a) Calculating the x-component of the acceleration of the center of mass:
a_cm = (F - f) / m
(b) Calculating the x-component of the friction force:
α = a_cm / r
τ = ICM * α
f = τ / r
Now we can substitute the values and calculate the answers.
To find the x-component of the acceleration of the center of mass of the yoyo (a) and the x-component of the friction force (f), we can use the equations of rotational motion and Newton's second law.
(a) The x-component of the acceleration of the center of mass of the yoyo can be found by considering the torques acting on the yoyo. Since the yoyo is rolling without slipping, the friction force provides the torque that causes the rotation. The torque τ = Fr, where F is the force applied to the string and r is the radius of the central spindle.
The torque τ is also related to the angular acceleration α and the moment of inertia ICM by τ = ICM * α.
Equating the two expressions for torque, we have Fr = ICM * α.
The linear acceleration a of the center of mass is related to the angular acceleration α by a = α * r.
Substituting the value of α from the torque equation into the linear acceleration equation, we get a = (Fr / ICM) * r.
Given F = 20 N, r = 0.225 m, and ICM = 0.045 kg m^2, we can calculate the x-component of the acceleration:
a = (20 N * 0.225 m) / (0.045 kg m^2) * 0.225 m = 20 m/s^2
Therefore, the x-component of the acceleration of the center of mass of the yoyo is 20 m/s^2.
(b) The x-component of the friction force can be found by applying Newton's second law. The net force acting on the yoyo in the x-direction is the sum of the force applied to the string (F) and the x-component of the friction force (f).
Using Newton's second law, ΣF = m * a, where ΣF is the net force and m is the mass of the yoyo.
In this case, the net force is given by ΣF = F - f.
Therefore, F - f = m * a.
Given F = 20 N, m = 1 kg, and a = 20 m/s^2 (as calculated in part (a)), we can solve for the x-component of the friction force:
20 N - f = 1 kg * 20 m/s^2
f = 20 N - 1 kg * 20 m/s^2
f = 20 N - 20 N
f = 0 N
Therefore, the x-component of the friction force is 0 N.