A point (labelled P in the figure) is attached to the rim of a disk of radius 0.2 m, which can turn around an axis through its center. It is rotating counterclockwise with a speed of 2.5 m/s.
Can someone please step me through the problem step by step?
Sure! Let's break down the problem step by step:
Step 1: Understand the problem
We have a disk with a point P attached to its rim. The disk can rotate around its center axis. The radius of the disk is 0.2 m, and it is rotating counterclockwise with a speed of 2.5 m/s. We need to solve the problem using step-by-step calculations.
Step 2: Draw a diagram
To better visualize the problem, try drawing a diagram. Draw a circle to represent the disk and label its center as O. Then, label the point attached to the rim as P.
O
\
\ P
Step 3: Identify what we want to find
The problem doesn't explicitly state what we need to find, so let's break it down. Since the point P is attached to the rim of the disk, it will undergo circular motion. We can use the equations of circular motion to solve for different quantities, such as the angular velocity, centripetal acceleration, or period of rotation. Let's solve for the following:
a) Angular velocity (ω)
b) Centripetal acceleration (a)
c) Period of rotation (T)
d) Tangential velocity (v)
Step 4: Solve for angular velocity (ω)
The angular velocity (ω) represents how fast an object rotates. It is defined as the change in angle per unit time.
In this problem, we are given the tangential velocity (2.5 m/s) and the radius (0.2 m). We can use the formula: ω = v/r.
ω = 2.5 m/s / 0.2 m
ω = 12.5 rad/s
So the angular velocity of the disk is 12.5 rad/s.
Step 5: Solve for centripetal acceleration (a)
Centripetal acceleration represents the acceleration towards the center of circular motion. It can be calculated using the formula: a = ω² * r.
a = (12.5 rad/s)² * 0.2 m
a = 156.25 m/s²
Therefore, the centripetal acceleration of the disk is 156.25 m/s².
Step 6: Solve for period of rotation (T)
The period of rotation (T) represents the amount of time it takes for the disk to complete one full revolution. It is defined as the reciprocal of the angular velocity.
We can use the formula: T = 1/ω.
T = 1 / 12.5 rad/s
T ≈ 0.08 seconds
The period of rotation of the disk is approximately 0.08 seconds.
Step 7: Solve for tangential velocity (v)
The tangential velocity (v) represents the velocity of an object moving in a circular path. It is given in the problem as 2.5 m/s.
Therefore, the tangential velocity of the point P is 2.5 m/s.
Step 8: Review the solution
To summarize, we have found the following quantities:
- Angular velocity (ω) = 12.5 rad/s
- Centripetal acceleration (a) = 156.25 m/s²
- Period of rotation (T) ≈ 0.08 seconds
- Tangential velocity (v) = 2.5 m/s
These steps should help you solve the problem step by step. Make sure to double-check your calculations and pay attention to units.