You have a 30g quarter on an old LP turntable at a radius of 30 cm from the turntable's center. If the coefficient of Friction between the coin and the turntable is 0.6, at what speed will the coin slide off the turntable? Hint: this is a circular motion problem. To find the necessary speed use the circumference of the circle the coin travels on.
Need Step by Step Solution.
When it starts the slide, the force of friction is just equal to centripetal force.
mass*angular speed *radius= mu*mass*g
solve for angular speed.
How would I convert that to m/s
To find the necessary speed for the coin to slide off the turntable, we need to consider the centripetal force acting on the coin and compare it to the force of friction.
Step 1: Determine the centripetal force.
The centripetal force is the force that keeps an object moving in a circular path. It is given by the equation:
F = m * a
where F is the centripetal force, m is the mass of the object, and a is the centripetal acceleration. In this case, the centripetal force is due to the tension in the turntable.
Step 2: Calculate the centripetal acceleration.
The centripetal acceleration of an object moving in a circle of radius r with a velocity v can be calculated using the equation:
a = v^2 / r
where a is the centripetal acceleration, v is the velocity, and r is the radius.
Step 3: Find the centripetal force.
We can use the equation from Step 1 to find F:
F = m * a = m * (v^2 / r)
Step 4: Determine the force of friction.
The force of friction, Ff, is given by the equation:
Ff = μ * N
where μ is the coefficient of friction and N is the normal force. In this case, the normal force is equal to the weight of the coin because it is resting on the turntable.
Step 5: Find the normal force.
The normal force, N, is equal to the weight of the coin, which is given by the equation:
N = m * g
where m is the mass of the coin and g is the acceleration due to gravity.
Step 6: Calculate the force of friction.
Using the equation from Step 4:
Ff = μ * N = μ * (m * g)
Step 7: Equate the centripetal force and the force of friction.
To find the speed at which the coin will slide off the turntable, we need to equate the centripetal force and the force of friction:
m * (v^2 / r) = μ * (m * g)
Step 8: Solve for v.
Solve the equation from Step 7 for v:
v = √((μ * r * g) / m)
Step 9: Plug in the values and calculate.
Substitute the given values into the equation from Step 8 and calculate the speed:
v = √((0.6 * 0.3 * 9.8) / 0.03)
v ≈ √(17.64 / 0.03)
v ≈ √588
v ≈ 24.2 cm/s
Therefore, the necessary speed for the coin to slide off the turntable is approximately 24.2 cm/s.