A record turntable is rotating at 33 and 1/3 rev/min. A watermelon seed is on the turntable 7.5 from the axis of rotation
Calculate the acceleration of the seed assuming that it does not slip?
Suppose that the turntable achieves its angular speed by starting from rest and undergoing a constant angular acceleration for .75 seconds. Calculate the minimum coefficient of static friction required for the seed not to slip during the acceleration period?
I do not know if it is 7.5 nautical miles or what, so I will call the distance from the center r.
convert 33 1/3 rev/mi to radians/s
w = 33 1/3 rev/min *1 min/60 s *2 pi rad/rev
= 3.49 rad/s
a = w^2 r = 12.2 r
in units of length you are using/s^2
since I do not know your length unit I must leave g as g instead of 9.8 m/s^2 or 32 ft/s^2 or whatever
mu = friction coef
Now we also have a tangential friction problem because the table is accelerating
tangential acceleration = r*(change in w/change in time)
= r * ( 3.49 rad/s /.75 s)
= r * 4.65 length units/s^2
total acceleration = sqrt(tangenial^2 + radial^2)
= r sqrt (4.65^2 + 12.2^2)
= r sqrt (170)
= 13.0 r
Now friction force = m a
m g mu = m (13 r)
mu = 13 r/g
To calculate the acceleration of the seed on the turntable, we can use the equation:
a = r * ω^2
where:
a is the acceleration,
r is the distance of the seed from the axis of rotation, and
ω is the angular velocity.
For the first question, we know that the turntable is rotating at 33 and 1/3 rev/min, which can be converted to angular velocity in radians per second. To do this, we need to multiply by 2π to convert revolutions to radians and divide by 60 to convert minutes to seconds:
ω = (33 and 1/3 rev/min) * (2π rad/rev) * (1 min/60 s)
ω = (33.333...) * (2π/60) rad/s
ω ≈ 3.49 rad/s
We are also given that the distance of the seed from the axis of rotation is 7.5 cm, which can be converted to meters:
r = 7.5 cm * (1 m / 100 cm)
r = 0.075 m
Now we can substitute these values into the equation to find the acceleration of the seed:
a = (0.075 m) * (3.49 rad/s)^2
a ≈ 0.897 m/s^2
Therefore, the acceleration of the seed assuming it does not slip is approximately 0.897 m/s^2.
For the second question, we need to calculate the minimum coefficient of static friction required for the seed not to slip during the acceleration period. The condition for the seed not to slip is that the frictional force between the seed and the turntable is equal to or greater than the maximum static frictional force. The maximum static frictional force can be calculated using the equation:
f_max = μ_s * N
where:
f_max is the maximum static frictional force,
μ_s is the coefficient of static friction, and
N is the normal force.
Since we are considering the seed on the turntable during the acceleration period, the normal force is equal to the weight of the seed, which can be calculated using the equation:
N = m * g
where:
m is the mass of the seed, and
g is the acceleration due to gravity.
We are not given the mass of the seed, so we cannot directly calculate the normal force. However, we know that the force required to accelerate an object is equal to the mass of the object multiplied by its acceleration:
F = m * a
We also know that at the moment of acceleration, the frictional force is equal to the force required to accelerate the seed:
f_max = F
Therefore, we can write:
μ_s * N = m * a
Substituting N = m * g, we get:
μ_s * (m * g) = m * a
Simplifying, we find:
μ_s = a / g
Now, we can substitute the value of acceleration (a) and the acceleration due to gravity (g) to find the minimum coefficient of static friction:
μ_s = 0.897 m/s^2 / 9.8 m/s^2
μ_s ≈ 0.092
Therefore, the minimum coefficient of static friction required for the seed not to slip during the acceleration period is approximately 0.092.
To calculate the acceleration of the seed, we need to find its tangential acceleration.
1. Convert the rotation speed from rev/min to rad/s:
33 and 1/3 rev/min = (33 + 1/3) rev/min = 33.333 rev/min
1 rev = 2π radians
33.333 rev/min = (33.333 x 2π) rad/min = (33.333 x 2π / 60) rad/s
= 3.49 rad/s
2. Calculate the tangential speed of the seed:
Tangential speed = distance from the axis of rotation * angular speed
Tangential speed = 7.5 m * 3.49 rad/s
= 26.175 m/s
3. Now, we can calculate the tangential acceleration:
Tangential acceleration = change in tangential speed / time
Since the seed is rotating at a constant speed, the change in tangential speed is 0.
Tangential acceleration = 0 / time
= 0 m/s^2
Therefore, the acceleration of the seed is 0 m/s^2. This means that the seed does not experience any acceleration.
To calculate the minimum coefficient of static friction required for the seed not to slip during the acceleration period, we need to consider rotational dynamics.
4. The angular acceleration of the turntable can be calculated using the equation:
angular acceleration = change in angular velocity / time
since the turntable starts from rest,
angular acceleration = final angular velocity / time
angular acceleration = 3.49 rad/s / 0.75 s
= 4.65 rad/s^2
5. The minimum coefficient of static friction can be calculated using the equation:
coefficient of static friction = (angular acceleration * distance from the axis of rotation) / radius of the turntable
coefficient of static friction = (4.65 rad/s^2 * 7.5 m) / r
where r is the radius of the turntable.
Note: The radius of the turntable is not provided in the question, so the specific value is required to calculate the minimum coefficient of static friction.