A 20g steel weight is placed on the edge of 1 30cm diameter plastic disk. The coefficient of static friction between the steel weight and plastic disk is 0.4. At what angular velocity will the weight slide off the disk?
Well, well, well! Looks like we have a classic case of a slippery situation. Let's see if we can uncover the answer with a touch of humor!
To find the angular velocity when the weight slides off the disk, we need to consider the balance between the gravitational force pulling downward and the friction force keeping it in place. But before we dive into the calculations, let me entertain you with a little joke:
Why don't scientists trust atoms?
Because they make up everything! 😄
Now, back to our problem. The first step is to calculate the static friction force, which can be found by multiplying the coefficient of static friction (0.4) with the normal force. The normal force is equal to the weight of the weight (20g) multiplied by the acceleration due to gravity (9.8 m/s²).
Next, we need to find the moment of inertia of the disk. For a solid disk, the moment of inertia (I) equals (1/2) * mass * radius^2.
Once we have these values, we can set up an equation of equilibrium to find the angular velocity at which the weight will slide off the disk.
But hey, here's another joke for you while we're at it:
Why did the scarecrow win an award?
Because he was outstanding in his field! 😂
Alright, let's crunch some numbers and get to the solution. Just give me a moment!
To determine the angular velocity at which the steel weight will slide off the plastic disk, we need to consider the forces acting on the weight.
The weight is about to slide, so the maximum static friction force is acting at the edge of the disk, opposing the impending motion. The maximum static friction force can be calculated using the equation:
F_max = μ_s * N
where F_max is the maximum static friction force, μ_s is the coefficient of static friction, and N is the normal force.
First, let's calculate the normal force acting on the weight. Since the weight is placed on the edge of the disk, it will experience a normal force equal to its weight. The weight can be calculated using the formula:
Weight = mass * gravitational acceleration
Given that the mass of the steel weight is 20g (0.02 kg) and the acceleration due to gravity is approximately 9.81 m/s^2, we can calculate the weight:
Weight = 0.02 kg * 9.81 m/s^2 = 0.1962 N
Now, we can calculate the maximum static friction force:
F_max = 0.4 * 0.1962 N = 0.07848 N
Next, let's calculate the torque acting on the weight. The torque can be calculated using the equation:
Torque = radius * force
The radius of the disk is half its diameter, which is 30 cm (0.3 m) / 2 = 0.15 m.
Torque = 0.15 m * 0.07848 N = 0.011772 Nm
Finally, let's calculate the angular velocity at which the weight will slide off the disk. The torque can be calculated using the equation:
Torque = moment of inertia * angular acceleration
The moment of inertia of a disk can be calculated using the formula:
Moment of inertia (disk) = (1/2) * mass_disk * radius_disk^2
Since the mass of the disk is not given, we need to assume it. Let's assume the mass of the disk is 100g (0.1 kg).
Moment of inertia (disk) = (1/2) * 0.1 kg * (0.15 m)^2 = 0.001125 kg.m^2
Now, we can calculate the angular acceleration:
angular acceleration = Torque / moment of inertia (disk) = 0.011772 Nm / 0.001125 kg.m^2 = 10.46 rad/s^2
Finally, we can calculate the angular velocity using the equation:
Final angular velocity^2 = Initial angular velocity^2 + 2 * angular acceleration * angular displacement
Since the initial angular velocity is 0, and we want to find the angular velocity at which the weight will slide off, we can set the initial angular velocity equal to 0:
Final angular velocity^2 = 0 + 2 * 10.46 rad/s^2 * angular displacement
Assuming the weight moves completely around the disk, the angular displacement can be calculated using the equation:
Angular displacement = 2 * π radians
Simplifying the equation for final angular velocity:
Final angular velocity^2 = 2 * 10.46 rad/s^2 * 2 * π radians
Final angular velocity^2 ≈ 131.95 rad^2/s^2
Taking the square root of both sides:
Final angular velocity ≈ √(131.95 rad^2/s^2)
Final angular velocity ≈ 11.49 rad/s
Therefore, the weight will slide off the disk when it reaches an angular velocity of approximately 11.49 rad/s.
To determine the angular velocity at which the weight will slide off the disk, we need to consider the balance of forces acting on the weight.
The force keeping the weight in place is the static friction force, which can be calculated using the equation:
Static friction force (fs) = coefficient of static friction (μs) * normal force (N)
The normal force, in this case, is the force exerted on the weight by the disk due to gravity and can be calculated using the equation:
Normal force (N) = mass (m) * gravitational acceleration (g)
Since the weight is placed on the edge of the disk, the radial distance between the center of the disk and the weight is equal to half the diameter of the disk, i.e., 15 cm (or 0.15 m).
In rotational motion, the centripetal force (Fc) acting on the weight is given by the equation:
Centripetal force (Fc) = mass (m) * radial distance (r) * angular velocity squared (ω^2)
For the weight to slide off the disk, the static friction force must be equal to or less than the maximum static friction force. Therefore, we need to determine the maximum static friction force (fs_max) using:
Maximum static friction force (fs_max) = coefficient of static friction (μs) * normal force (N)
Now, setting the value of fs_max equal to Fc, we can solve for the angular velocity (ω).
Let's calculate it step by step:
1. Calculate the normal force (N):
N = m * g
N = 0.02 kg * 9.8 m/s²
N = 0.196 N (to three significant figures)
2. Calculate the maximum static friction force (fs_max):
fs_max = μs * N
fs_max = 0.4 * 0.196 N
fs_max = 0.0784 N (to four significant figures)
3. Equate fs_max and Fc:
fs_max = Fc
0.0784 N = 0.02 kg * 0.15 m * ω²
0.0784 N = 0.003 kg * 0.15 m * ω²
4. Solve for ω:
ω² = (0.0784 N) / (0.003 kg * 0.15 m)
ω² = 174.22 (to five significant figures)
ω ≈ √174.22
ω ≈ 13.18 rad/s (to two significant figures)
Therefore, the weight will slide off the disk when it reaches an angular velocity of approximately 13.18 radians per second.