A yo-yo of mass m rests on the floor (the static friction coefficient with the floor is mu ). The inner (shaded) portion of the yo-yo has a radius R-1 , the two outer disks have radii R-2 . A string is wrapped around the inner part. Someone pulls on the string at an angle Beta (see sketch). The "pull" is very gentle, and is carefully increased until the yo-yo starts to roll without slipping.
For what angles of beta will the yo-yo roll to the left and for what angles to the right?
a) Yo-Yo rolls to the left if sin beta<R-1/R_2 , and to the right if sin beta> R_1/R_2.
b)Yo-Yo rolls to the left if sin beta>R-1/R_2 , and to the right if sin beta<R_1/R_2
c)yo- yo rolls to the left if cos beta< R_1/R_2 and to the right if cos beta>R_1/R-2
d)yo yo rolls to the left if cosbeta>R_1/R_2 and to the right if cosbeta< R_1/R_2 . .
To determine the direction in which the yo-yo will roll, we need to analyze the forces acting on it.
When the yo-yo starts to roll without slipping, there are two important factors to consider:
1. Torque: The torque exerted by the tension in the string must be sufficient to overcome the static friction between the yo-yo and the floor. This ensures that the yo-yo starts rolling without slipping.
2. Friction: The direction of the friction force will determine the direction in which the yo-yo rolls.
Let's analyze the forces acting on the yo-yo when it starts rolling without slipping:
1. Tension in the string: The tension in the string exerts a torque on the yo-yo. This torque is given by τ = T * R₁, where T is the tension in the string and R₁ is the radius of the shaded inner portion of the yo-yo.
2. Friction force: The friction force between the yo-yo and the floor will oppose the torque exerted by the tension in the string. It is given by f = μ * N, where μ is the static friction coefficient and N is the normal force exerted by the floor on the yo-yo.
Since the yo-yo is not accelerating in the vertical direction, we can balance the forces in that direction:
N - mg = 0, where m is the mass of the yo-yo, and g is the acceleration due to gravity.
Solving for N, we get N = mg.
Now, to determine the direction of the yo-yo's motion, we need to compare the torque exerted by the tension in the string (τ) with the torque opposing it due to the friction force (f):
If τ > f, the yo-yo will roll to the left.
If τ < f, the yo-yo will roll to the right.
We know that τ = T * R₁ and f = μ * N = μ * mg.
Substituting these values and simplifying, we get:
If T * R₁ > μ * mg, the yo-yo will roll to the left.
If T * R₁ < μ * mg, the yo-yo will roll to the right.
Now, let's rearrange the inequality to express tension (T) in terms of other variables:
T > μ * m * g / R₁ for the yo-yo to roll to the left.
T < μ * m * g / R₁ for the yo-yo to roll to the right.
Now, let's consider the angle β at which the string is pulled:
sin β = R₁ / R₂, where R₂ is the radius of the two outer disks of the yo-yo.
From this equation, we can rearrange to express R₁ in terms of R₂:
R₁ = R₂ * sin β
Substituting this expression for R₁ in the equations for tension, we get:
If T > μ * m * g / (R₂ * sin β), the yo-yo will roll to the left.
If T < μ * m * g / (R₂ * sin β), the yo-yo will roll to the right.
Notice that in the expression for the tension on both sides, the variables μ, m, g, and R₂ are constant.
Now, comparing these expressions to the provided answer options:
a) Yo-yo rolls to the left if sin β < R₁ / R₂, and to the right if sin β > R₁ / R₂.
This is incorrect because it does not match the derived expressions.
b) Yo-yo rolls to the left if sin β > R₁ / R₂, and to the right if sin β < R₁ / R₂.
This is correct based on the derived expressions.
c) Yo-yo rolls to the left if cos β < R₁ / R₂, and to the right if cos β > R₁ / R₂.
This is incorrect because it does not match the derived expressions.
d) Yo-yo rolls to the left if cos β > R₁ / R₂, and to the right if cos β < R₁ / R₂.
This is incorrect because it does not match the derived expressions.
Therefore, the correct answer is b) "Yo-yo rolls to the left if sin β > R₁ / R₂, and to the right if sin β < R₁ / R₂."