A 28 g block sits at the center of a turntable that rotates at 60 rpm. A compressed spring shoots the block radially outward from the center along a frictionless groove in the surface of the turntable. Calculate the turntable's angular speed when the block reaches the outer edge. Treat the turntable as a solid disk with mass with mass 200 g and diameter 44.0 cm. Express your answer in revolutions per minute.
final momentum=initialmomentum
final momentkum= 1/2 (.2*.44)w+.028*w
initial momentum= 1/2 (.2*.44)60
set them equal, solve for w. Notice at the center, r=0, so the block has no angular momentum.
To calculate the turntable's angular speed when the block reaches the outer edge, we can use the principle of conservation of angular momentum.
The initial angular momentum of the system is equal to the final angular momentum of the system. We can express this as:
Initial Angular Momentum = Final Angular Momentum
The initial angular momentum is given by the block's initial rotational motion at the center of the turntable. Since the block is at rest initially, its angular momentum is zero.
The final angular momentum is the sum of the block's angular momentum when it reaches the outer edge and the turntable's angular momentum.
The angular momentum of the block when it reaches the outer edge can be calculated using the equation:
Angular Momentum of the block = (Moment of inertia of the block) * (Angular velocity of the block)
The moment of inertia of the block can be calculated using the equation:
Moment of inertia of a solid disk = (1/2) * (mass of the disk) * (radius of the disk)^2
The angular velocity of the block can be calculated using the equation:
Angular velocity of the block = (velocity of the block) / (radius of the disk)
The velocity of the block can be calculated using energy conservation. The initial potential energy of the spring is converted into kinetic energy when the block reaches the outer edge. Therefore, we can use the equation:
Initial Potential Energy of the spring = Kinetic Energy of the block at the outer edge
The potential energy of the spring can be calculated using Hooke's law:
Potential Energy of the spring = (1/2) * (spring constant) * (spring displacement)^2
The spring constant can be assumed to be k = 60 N/m.
The spring displacement can be calculated using the equation:
Spring displacement = (radius of the disk) - (radius of the block)
The kinetic energy of the block can be calculated using the equation:
Kinetic Energy of the block at the outer edge = (1/2) * (mass of the block) * (velocity of the block)^2
We can substitute the equations for the potential energy of the spring and the kinetic energy of the block into each other to solve for the velocity of the block.
Finally, we can substitute the velocity of the block into the equation for the angular velocity of the block to obtain the final angular momentum.
Once we have the final angular momentum, we can calculate the turntable's angular velocity (angular speed) using the equation:
Angular velocity of the turntable = (final angular momentum) / (moment of inertia of the turntable)
The moment of inertia of the turntable can be calculated using the equation for the moment of inertia of a solid disk.
Finally, the angular speed can be expressed in revolutions per minute by converting the angular speed to revolutions per second and multiplying by 60.
Let's calculate the turntable's angular speed step by step.
To calculate the turntable's angular speed when the block reaches the outer edge, we need to apply the principle of conservation of angular momentum.
First, we need to calculate the initial angular momentum of the system. Since the block is initially at the center of the turntable, its initial angular momentum is zero.
The angular momentum of the turntable can be calculated using the formula:
L = I * ω
where L is the angular momentum, I is the moment of inertia, and ω is the angular speed.
The moment of inertia of a solid disk can be calculated using the formula:
I = (1/2) * m * r^2
where I is the moment of inertia, m is the mass of the disk, and r is the radius of the disk.
In this case, the mass of the disk is given as 200 g, which is equivalent to 0.2 kg. The radius of the disk can be calculated as half of the diameter:
r = 0.5 * diameter = 0.5 * 44.0 cm = 22.0 cm = 0.22 m
Plugging in these values, we can calculate the moment of inertia of the disk:
I = (1/2) * 0.2 kg * (0.22 m)^2 = 0.00242 kgm^2
Since the initial angular momentum is zero, we can write:
0 = I * ω_initial
Solving for ω_initial, we find:
ω_initial = 0
When the block reaches the outer edge, its radius is equal to the radius of the disk. We'll denote this radius as R.
The final angular momentum of the system can be calculated as:
L_final = (m_block * R^2 + I) * ω_final
where m_block is the mass of the block.
In this case, the mass of the block is given as 28 g, which is equivalent to 0.028 kg. The radius of the turntable can be calculated as the radius of the disk plus the radius of the block:
R = r + r_block
where r_block is the radius of the block.
Since the block reaches the outer edge, we can write:
R = r + r_block
Solving for r_block, we find:
r_block = R - r
Plugging in the values of r and R, we can calculate the radius of the block:
r_block = 0.22 m - 0.028 m = 0.192 m
Now we can calculate the final angular momentum of the system:
L_final = (0.028 kg * (0.192 m)^2 + 0.00242 kgm^2) * ω_final
Since angular momentum is conserved, we know that the initial and final angular momentum are equal. Therefore:
I * ω_initial = (0.028 kg * (0.192 m)^2 + 0.00242 kgm^2) * ω_final
Solving for ω_final, we find:
ω_final = (I * ω_initial) / (0.028 kg * (0.192 m)^2 + 0.00242 kgm^2)
Substituting the values of I and ω_initial, we can calculate ω_final:
ω_final = (0 * 0) / (0.028 kg * (0.192 m)^2 + 0.00242 kgm^2) = 0
Therefore, the turntable's angular speed when the block reaches the outer edge is 0 revolutions per minute.