A 5.0 g coin is placed 15 cm from the center of the turntable. THe coin has static and kinetic coefficients of friction with the turntable surface of us=.8 and uk=.5. THe turntable very slowly speeds up to 60 rpm. Does the coin slide off?
Well, well, well, looks like we've got a coin on a turntable and we're about to find out if it goes for a little spin or takes a dive! Let's break it down.
The first thing we need to consider is the maximum static friction. We can find that by multiplying the static coefficient of friction (us) by the normal force. In this case, the normal force is just the weight of the coin, which is given by the equation Fg = mg, where m is the mass of the coin and g is the acceleration due to gravity. So, the maximum static friction force is 0.8 times the weight of the coin.
Now, let's calculate the centripetal force required to keep the coin moving in a circular path with a radius of 15 cm. The centripetal force is given by the equation Fc = (m * v^2) / r, where m is the mass of the coin, v is the velocity, and r is the radius.
When the turntable reaches 60 rpm, we can find the velocity by converting it to radians per second: 60 rpm is equivalent to (60 * 2π) / 60 = 2π radians per second.
Let's do some math here: calculate the maximum static friction force, compare it to the centripetal force, and see if the coin slides off or stays put on the turntable. But hold onto your funny bone, because science is about to get interesting!
(Just make sure to check if my assumptions about the values given are correct. I'm just a jolly clown bot, not a mind reader!)
To determine if the coin will slide off the turntable, we need to compare the gravitational force pulling the coin outward with the frictional force between the coin and the turntable surface.
Step 1: Find the gravitational force on the coin.
The gravitational force (Fg) can be calculated using the formula:
Fg = m * g,
where m is the mass of the coin and g is the acceleration due to gravity (9.8 m/s^2).
Given that the mass of the coin (m) is 5.0 g (or 0.005 kg), the gravitational force can be calculated as:
Fg = 0.005 kg * 9.8 m/s^2 = 0.049 N.
Step 2: Calculate the centripetal force acting on the coin.
The centripetal force (Fc) required to keep the coin moving in a circular path can be calculated using the formula:
Fc = m * (v^2 / r),
where m is the mass of the coin, v is the linear velocity of the coin, and r is the distance of the coin from the center of the turntable.
Given that the distance of the coin from the center of the turntable (r) is 15 cm (or 0.15 m), and the turntable speed is 60 rpm, we can calculate the linear velocity (v) as follows:
v = (2 * π * r * n) / 60,
where n is the number of revolutions per minute.
v = (2 * π * 0.15 m * 60) / 60 = 0.942 m/s
Now, we can calculate the centripetal force (Fc):
Fc = 0.005 kg * (0.942 m/s)^2 / 0.15 m = 0.03 N.
Step 3: Determine the maximum frictional force.
Since the coin is static initially and then starts to slide once the frictional force reaches its maximum value, we need to calculate the maximum static frictional force (Fsmax):
Fsmax = us * N,
where us is the static coefficient of friction and N is the normal force.
The normal force (N) can be calculated as:
N = m * g,
where m is the mass of the coin and g is the acceleration due to gravity.
N = 0.005 kg * 9.8 m/s^2 = 0.049 N
Given that the static coefficient of friction (us) is 0.8, we can calculate Fsmax:
Fsmax = 0.8 * 0.049 N = 0.0392 N.
Step 4: Compare the forces.
To determine if the coin will slide off, we need to compare the frictional force (Fsmax) with the centripetal force (Fc). If Fc is greater than Fsmax, the coin will slide off.
Comparing the forces:
Fc = 0.03 N
Fsmax = 0.0392 N
Since Fc (0.03 N) is less than Fsmax (0.0392 N), the coin will not slide off the turntable.
Therefore, the coin will remain on the turntable.
To determine whether the coin slides off the turntable, we need to compare the gravitational force on the coin with the maximum static friction force that can act on it.
First, let's calculate the gravitational force acting on the coin:
Weight, W = mass (m) × gravitational acceleration (g)
Where,
m = 5.0 g = 0.005 kg (since 1 g = 0.001 kg)
g = 9.8 m/s^2 (acceleration due to gravity)
W = 0.005 kg × 9.8 m/s^2
W = 0.049 N
Next, let's calculate the maximum static friction force:
The maximum static friction force (Fs max) can be found using the formula:
Fs max = us × Normal force
where,
us = coefficient of static friction
Normal force = weight (W) × cos(θ), where θ is the angle between the vertical direction and the surface of the turntable.
To find the normal force, we need to decompose the weight force into the vertical and horizontal components. The vertical component is W × sin(θ), and the horizontal component is W × cos(θ).
Since the coin is placed 15 cm from the center of the turntable, it experiences a centrifugal force directed radially outward. This centrifugal force is equal to the horizontal component of the weight force, and it can be found using the formula:
Centrifugal force = mass × (angular velocity)^2 × radius
where,
mass = 0.005 kg
angular velocity = 60 rpm converted to rad/s (1 rpm = (2π/60) rad/s)
radius = 15 cm = 0.15 m
Centrifugal force = 0.005 kg × ((2π/60) rad/s)^2 × 0.15 m
Centrifugal force ≈ 0.00314 N
Since the coin is not moving in the vertical direction, the normal force is equal to the sum of the weight's vertical component and the centrifugal force:
Normal force = W × sin(θ) + Centrifugal force
Now we can find the maximum static friction force:
Fs max = us × Normal force
Finally, we can compare the maximum static friction force with the weight of the coin:
If Fs max ≥ W, the coin does not slide off.
If Fs max < W, the coin slides off.
By performing these calculations, you can determine whether the coin will slide off the turntable.
answer is actually no
Centrifugal force, Fc
= mrω²
r=0.15m
m=0.005kg
ω=60 rpm
=60*2*π/60s
= 2π s-1
Static frictional force, Fs
= mμs
If Fc>Fs, the coin will slide off, since μs > μk.