1.If a mass is 15 cm from the axis of rotation and makes 25 revolutions in 10 seconds, what is its speed in m/s?
2.If m = 0.1 kg and the values above still hold, what is the numerical values of the centripetal force?
1. 25 rev/10 s = 2.5 rev/s
w = 2.5 *2*pi = 15.71 radians/s.
Multiply that by the radius for the speed.
V = 235.6 cm/s = 2.356 m/s
2. centripetal force = , m V^2/r
= m r w^2
1. Well, let's break it down. If there are 25 revolutions in 10 seconds, that means there are 2.5 revolutions per second. And since the circumference of a circle is 2πr, the distance covered in one revolution is 2π times the radius. So, the distance covered in one second is 2.5 times 2π times the radius. But the radius is 15cm, which is 0.15m. So, the distance covered in one second is 2.5 times 2π times 0.15m. And finally, since speed is distance over time, the speed is (2.5 times 2π times 0.15m) divided by 1 second. And when you do the math, you get... drum roll please... the speed is approximately 2.356 meters per second!
2. Ah, the numerical value of the centripetal force. Well, the centripetal force is given by the equation F = mω²r, where F is the centripetal force, m is the mass, ω is the angular velocity, and r is the radius. In this case, you already know the mass is 0.1kg, the angular velocity is 2π times 2.5 (since there are 2.5 revolutions per second), and the radius is 0.15m. So, just plug those values into the equation and calculate. And voilà, you'll have the numerical value of the centripetal force!
To answer these questions, we will use the following formulas:
1. Speed (v) = 2πr/T
2. Centripetal force (F) = mv^2 / r
First, let's solve question 1:
Given:
- Distance from the axis of rotation (r) = 15 cm = 0.15 m
- Number of revolutions (n) = 25
- Time (T) = 10 seconds
We can calculate the speed using the formula:
v = 2πr/T
Plugging in the values:
v = (2π * 0.15 m)/(10 s)
Calculating the value:
v ≈ 0.094 m/s
Therefore, the speed of the mass is approximately 0.094 m/s.
Moving on to question 2:
Given:
- Mass (m) = 0.1 kg
- Speed (v) = 0.094 m/s
- Distance from the axis of rotation (r) = 0.15 m
We can calculate the centripetal force using the formula:
F = mv^2 / r
Plugging in the values:
F = (0.1 kg * (0.094 m/s)^2) / 0.15 m
Calculating the value:
F ≈ 0.0626 N
Therefore, the numerical value of the centripetal force is approximately 0.0626 N.
To find the speed of the mass in question 1, we can use the formula for linear speed:
Linear Speed = (2πr * number of revolutions) / time
First, we convert the given distance from cm to meters:
15 cm = 0.15 m
Next, we calculate the linear speed:
Linear Speed = (2π * 0.15 * 25) / 10
Simplifying this equation, we have:
Linear Speed = (3.14 * 0.15 * 25) / 10
Linear Speed = 1.1775 m/s
Therefore, the speed of the mass is approximately 1.1775 m/s.
Moving on to question 2, we are now asked to calculate the numerical value of the centripetal force. The formula for centripetal force is:
Centripetal Force = mass * (linear speed)^2 / radius
Given data:
Mass (m) = 0.1 kg
Linear Speed = 1.1775 m/s
Radius (r) = 0.15 m
Now we can substitute these values into the formula:
Centripetal Force = 0.1 kg * (1.1775 m/s)^2 / 0.15 m
Simplifying this equation, we have:
Centripetal Force = 0.1 kg * 1.38507625 m^2/s^2 / 0.15 m
Centripetal Force = 0.0138507625 kg*m/s^2 / 0.15 m
Centripetal Force = 0.0923384167 kg*m/s^2
Therefore, the numerical value of the centripetal force is approximately 0.0923 kg*m/s^2.