A mouse sits on a turntable 14.8 cm from the center. The coefficient of static friction is 0.49. The angular velocity increases steadily from zero. At what angular velocity does the mouse begin to slide?
I believe that the max frictional force can be is momentum x normal force = mass x velocity^2/radius.
I am just unsure how figure out how to get momentum.
To determine the angular velocity at which the mouse begins to slide, we need to compare the centripetal force acting on the mouse with the maximum static frictional force.
First, we can calculate the centripetal force acting on the mouse using the formula:
Centripetal force = mass x (angular velocity)^2 x radius
Where:
- Mass is the mass of the mouse
- Angular velocity is the angular velocity at which the mouse is rotating
- Radius is the distance from the center of the turntable to the mouse
Now, let's calculate the centripetal force acting on the mouse.
Next, we can calculate the maximum static frictional force using the formula:
Maximum static frictional force = coefficient of static friction x normal force
Where:
- Coefficient of static friction is the given value of 0.49
- Normal force is the force exerted by the turntable perpendicular to the surface of the turntable. In this case, it is equal to the weight of the mouse.
Finally, we can set the centripetal force equal to the maximum static frictional force and solve for the angular velocity at which the mouse begins to slide.
Centripetal force = Maximum static frictional force
mass x (angular velocity)^2 x radius = coefficient of static friction x weight
Now, let's substitute the given values and solve for the angular velocity.
To determine the angular velocity at which the mouse begins to slide, we can use the equation for the maximum static friction force:
fs max = μs * N
Where:
- fs max is the maximum static friction force
- μs is the coefficient of static friction
- N is the normal force
In this case, the normal force N is equal to the weight of the mouse since it is on a horizontal turntable. So we have:
N = mg
Where:
- m is the mass of the mouse
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
Now, to find the momentum, we'll need to express it in terms of angular velocity. The angular velocity (ω) of an object rotating in a circle can be related to its linear velocity (v) by the equation:
v = ω * r
Where:
- v is the linear velocity
- r is the distance from the center of rotation (in this case, 14.8 cm or 0.148 m)
The mass of the mouse (m) is given, so we can calculate its linear velocity using the equation:
v = ω * r
And because linear velocity (v) is related to momentum (p) through the equation:
p = m * v
We can now calculate the momentum (p) of the mouse.
To summarize the steps:
1. Calculate the normal force N = mg (where m is the mass of the mouse and g is the acceleration due to gravity).
2. Calculate the linear velocity v = ω * r (where r is the distance from the center of rotation and ω is the angular velocity).
3. Calculate the momentum p = m * v.
Once you have the momentum (p), you can substitute it into the equation for maximum static friction force:
fs max = μs * N
where μs is given in the problem.
The angular velocity at which the mouse begins to slide can be found by rearranging the equation:
ω = fs max / (m * r)
And substitute the values you have to find the answer.