A yoyo of mass m=2 kg and moment of inertia ICM=0.04 kg m2 consists of two solid disks of radius R=0.2 m, connected by a central spindle of radius r=0.15 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=12 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=
unanswered
(b) What is the x-component of the friction force? (in N)
f=
a=F.(R+r)/(I/R+m.R)=7
f=F-m.a =-2 BUT you write 2 (without the minus (-) sing)
Shouldn't the acceleration be less than 6 m/s^2? The yoyo should not accelerate more than what is provided by the 12N force or a(max)= F/m = 12/2 = 6 m/s^2 ?
To solve this problem, let's break it down step by step:
Step 1: Find the torque generated by the applied force:
Since the yoyo is rolling without slipping, the applied force F will cause a torque.
The torque (τ) is given by the formula: τ = Iα
Here, I is the moment of inertia of the yoyo (ICM) and α is the angular acceleration of the yoyo.
Given ICM = 0.04 kg m^2, we can use this value to find the torque.
Step 2: Find the angular acceleration:
To find the angular acceleration, we can use Newton's second law for rotation:
τ = Iα
Where τ is the torque and α is the angular acceleration.
Since the torque (τ) is caused by the applied force F and the radius r, we can write:
τ = F * r
Substituting the values, τ = 12 N * 0.15 m = 1.8 Nm
Now we can solve for α:
1.8 Nm = 0.04 kg m^2 * α
α = (1.8 Nm) / (0.04 kg m^2) = 45 rad/s^2
Step 3: Find the linear acceleration:
The linear acceleration (a) of the center of mass of the yoyo can be found using the formula:
a = α * R
Where α is the angular acceleration and R is the radius of the yoyo.
Given α = 45 rad/s^2 and R = 0.2 m, we can calculate the linear acceleration:
a = 45 rad/s^2 * 0.2 m = 9 m/s^2
So, the x-component of the acceleration of the center of mass of the yoyo is 9 m/s^2.
Answer to (a): a = 9 m/s^2
Step 4: Find the x-component of the friction force:
The friction force (f) can be calculated using the equation:
f = μ * N
Where μ is the coefficient of friction and N is the normal force.
In this case, since the yoyo is placed on a flat rough surface, there is no vertical acceleration, and the normal force (N) is equal to the weight (mg) of the yoyo.
Given the mass m = 2 kg, we can calculate the normal force:
N = mg = 2 kg * 9.8 m/s^2 = 19.6 N
Now we need to find the coefficient of friction (μ). Since the question doesn't provide this information, we cannot determine the value of μ and solve for the friction force (f).
Answer to (b): The x-component of the friction force (f) cannot be determined without the coefficient of friction.
So, the answer to (b) is unknown without additional information.