A penny of mass 3.1g rests on a small 28.2g block supported by a spinning disk of radius 12cm. The coefficients of friction between the block and disk are .73 (static) and .64 (kinetic) while those for the penny and black are .6 (static) and .45 (kinetic). What is the maximum speed of the disk without the block and penny sliding?
Answer in units of rpm
67
To find the maximum speed of the disk without the block and penny sliding, we need to determine the conditions under which the frictional forces on both the block and the penny are maximized. Let's break down the problem and solve it step by step.
1. Calculate the force of gravity acting on the block and the penny:
- The force of gravity (Fg1) on the block is given by Fg1 = mass1 * g, where mass1 is the mass of the block and g is the acceleration due to gravity.
- Similarly, the force of gravity (Fg2) on the penny is given by Fg2 = mass2 * g, where mass2 is the mass of the penny.
2. Determine the maximum frictional force between the block and the disk:
- The maximum static friction (Fs) between the block and the disk can be calculated using the equation Fs = coefficient_static * Normal force, where coefficient_static is the coefficient of static friction between the block and the disk, and Normal force is the force perpendicular to the contact surface (equal to Fg1).
- Normal force (N) on the block is equal to Fg1.
3. Determine the maximum frictional force between the penny and the disk:
- The maximum static friction (Fs') between the penny and the disk can be calculated using the equation Fs' = coefficient_static * Normal force', where coefficient_static is the coefficient of static friction between the penny and the disk, and Normal force' is the force perpendicular to the contact surface (equal to Fg2).
- Normal force' (N') on the penny is equal to Fg2.
4. Calculate the maximum torque that the frictional forces on the block and penny can produce on the disk:
- The torque (τ) exerted by a force (F) on the disk at a distance (r) from the center of rotation is given by τ = F * r. In this case, F represents the frictional force on either the block or the penny, and r is the radius of the disk.
5. Find the maximum angular acceleration of the disk:
- The maximum angular acceleration (α) of the disk is given by α = τ / moment of inertia, where τ is the maximum torque exerted by the block and penny combined, and moment of inertia (I) is a property of the disk that depends on its mass distribution and shape.
6. Calculate the maximum rotational speed of the disk:
- The maximum rotational speed (ω) of the disk can be obtained by using the equation ω = √(2 * α * θ), where α is the maximum angular acceleration of the disk, and θ is the angle through which the disk rotates.
7. Convert the rotational speed from radians per second to revolutions per minute (rpm):
- To convert the rotational speed from radians per second to revolutions per minute, we need to multiply the angular speed by a conversion factor. One revolution is equal to 2π radians, and there are 60 seconds in a minute.
By following these steps, you should be able to find the maximum speed of the disk without the block and penny sliding.