Solving Rotational Motion Problem: A .005 kg Nickel

[wizcreations]

I've been having difficulties solving this problem. I will say how I have been trying to solve it beneath the question.


A .005 kg nickel rests .100 m from the center of a record player at rest. The nickel and player have a coefficient of friction of .33.
a) If the player can accelerate at 2.50 rad/s^. How many revolutions will it go through before the nickel is thrown off?
b) The player is level .985 m from the ground. Find the nickel's displacement from leaving the player to hitting the ground.


To solve this, I figured I need to find out what force is required to overcome the friction, so I found the nickel's weight as .005kg*9.80m/s^2 = .049N and then .049N*.33 = .016 N required for the nickel to be thrown off. From here, I wasn't exactly sure what to do. I'm pretty sure I need to find out what velocity is required to get that force, so I used acceleration=force/mass to get 3.2 m/s^2 as the acceleration. To get the velocity, I used centripetal acceleration = velocity^2 / radius and got .57 m/s. I have no idea if this is correct, and if it is, I don't know how to use the 2.50 rad/s^2 acceleration of the record player to see how long it takes to reach that velocity. Once I have the time requred to reach that speed, I would just find out how many revolutions it would make in that time.

For part b) I need to know both distance and direction. I'm pretty sure I can get distance on my own, but I don't know about direction.


Thank you for the help.

 

[AvroArrow]

For part a), the first steps you took are correct. After finding the force required to overcome the friction, you need to use the acceleration of the record player to find the time it takes to reach the velocity that will cause the nickel to be thrown off. This can be done using the equation v = at (where v is the final velocity, a is the acceleration of the record player, and t is the time it takes to get to that velocity). Since you know the acceleration and the final velocity (0.57 m/s), you can solve for t. Once you have the time, you can calculate the number of revolutions it goes through before the nickel is thrown off using the equation rev = ωt (where rev is the number of revolutions, ω is the angular velocity of the record player, and t is the time). For part b), to find the displacement of the nickel, you need to use the equation x = x0 + v0*t + 1/2*a*t^2 (where x is the displacement, x0 is the initial position, v0 is the initial velocity, t is the time the nickel spends in the air, and a is the acceleration due to gravity). Since you know the initial position and the acceleration due to gravity, you can solve for t by rearranging the equation to t = 2(x - x0)/(v0 + a*t). Afterwards, you can plug t into the original equation to find the displacement.

 

[m2003]

To solve this problem, you are on the right track by finding the force required to overcome the friction between the nickel and the record player. However, instead of using the nickel's weight, you should use the centripetal force formula, which is F=mv^2/r. In this case, the force required would be (0.005 kg)(2.50 rad/s)^2/(0.100 m) = 0.125 N. This is the minimum force required to keep the nickel in place on the record player.

To find the velocity at which the nickel will be thrown off, you can use the equation v=sqrt(u*r*a), where u is the coefficient of friction, r is the radius, and a is the acceleration. In this case, the velocity would be sqrt(0.33*0.100 m*2.50 rad/s^2) = 0.57 m/s. This means that if the record player accelerates at a rate of 2.50 rad/s^2, it will take approximately 0.23 seconds for the nickel to be thrown off. To find the number of revolutions, you can use the formula v=wr, where w is the angular velocity (2.50 rad/s) and r is the radius (0.100 m). This gives you a velocity of 0.25 m/s, which means the record player will make approximately 0.57/0.25 = 2.28 revolutions before the nickel is thrown off.

For part b), you can use the equation s=ut+1/2at^2, where s is the displacement, u is the initial velocity, a is the acceleration, and t is the time. In this case, the initial velocity is 0.57 m/s, the acceleration is -9.80 m/s^2 (due to gravity), and the time is 0.23 seconds. This gives a displacement of approximately 0.065 m. As for the direction, assuming the record player is rotating counterclockwise, the nickel will be thrown off in the direction opposite to the rotation, which is to the left.

I hope this helps you solve the problem. Remember to always use the appropriate formulas and units in your calculations. Good luck!