A 300 g block on a 54.0 cm -long string swings in a circle on a horizontal, frictionless table at 85.0 rpm .
What is the speed of the block?
Express your answer with the appropriate units.
What is the tension in the string?
Express your answer with the appropriate units.
angular velocity=w= 85r/min*1min/60sec*2PIrad/rev
= 85*2PI/60
speed=angularvelocty*radius
tension= mass*w^2*r
Well, well, well, let's dive into the swinging madness, shall we?
To find the speed of the block, we need to convert the rotation rate from rpm to radians per second (because we're fancy like that). One revolution is equal to 2π radians, so 85.0 rpm is equal to (85.0 revolution/minute) * (2π radians/1 revolution) * (1 minute/60 seconds). Crunching those numbers, we find that the rotation rate is approximately 8.90 radians per second.
Now, my friend, we can use the formula for the speed of an object in uniform circular motion: v = rω, where v is the speed, r is the radius of the circle, and ω is the angular velocity (in this case, 8.90 radians per second). We know the radius is half the length of the string, so r = 54.0 cm / 2 = 27.0 cm.
But hang on a second, we want the speed in meters per second because we're all about that metric system. So, we convert that pesky centimeters to meters: 27.0 cm * (1 m/100 cm) = 0.27 m.
Now, we just plug our numbers into the formula. Speed of the block = (0.27 m) * (8.90 radians/second) ≈ 2.40 m/s.
Voilà! The speed of the block is approximately 2.40 meters per second. Now, onto the tension in the string!
To find the tension in the string, we need to use some good ol' Newton's second law. The net force acting on the block is the centripetal force, which is equal to the tension in the string. This force can be calculated using the formula: F = m*v^2 / r, where F is the centripetal force, m is the mass of the block (300 g = 0.3 kg), v is the speed of the block (2.40 m/s), and r is, once again, the radius of the circle (0.27 m).
Let's plug in those numbers and solve for F: F = (0.3 kg) * (2.40 m/s)^2 / 0.27 m. Punching those numbers into the calculator, we find that the tension in the string is approximately 8.00 N.
So, my friend, the tension in the string is approximately 8.00 Newtons. Keep swinging and stay balanced out there!
To find the speed of the block, we can use the formula:
V = ω * r
Where:
- V is the speed of the block,
- ω (omega) is the angular velocity in radians per second, and
- r is the radius of the circular path.
To convert the given angular velocity from rpm to radians per second, we use the conversion factor:
1 revolution = 2π radians
First, we need to find the angular velocity in radians per second:
ω = (85.0 rpm) * 2π radians/1 revolution * 1 minute/60 seconds
Now, let's calculate the value:
ω = (85.0 * 2 * π) / 60 radians/second
Next, we can find the radius of the circular path by converting the given length of the string to meters:
r = (54.0 cm) / 100 cm/m
Now, we can substitute the values into the equation to find the speed of the block:
V = ω * r
Finally, we can calculate the answer.
To find the speed of the block, we can use the formula for the linear velocity of an object moving in a circular path:
v = ω * r
Where:
v = linear velocity
ω = angular velocity (in radians per second)
r = radius
First, let's convert the given angular velocity from rpm to radians per second. To do this, we need to use the conversion factor: 1 revolution = 2π radians.
Angular velocity (ω) = 85.0 rpm * (2π radians/1 revolution) * (1 minute/60 seconds)
= 85.0 * 2π * (1/60) radians/second
Next, we need to determine the radius of the circular path. We are given the length of the string, which is 54.0 cm. However, the length of the string is equal to the circumference of the circular path, so we need to divide it by 2π to get the radius:
radius (r) = 54.0 cm / (2π) cm/radian
Now we have all the values we need to calculate the speed of the block:
v = ω * r
Substituting in the values:
v = (85.0 * 2π * (1/60)) radians/second * (54.0 cm / (2π) cm/radian)
Simplifying the expression:
v = 85.0 * 54.0 / 60 cm/second
Therefore, the speed of the block is 76.5 cm/second.
Now let's calculate the tension in the string. The tension in the string is equal to the centripetal force acting on the block. The centripetal force can be calculated using the formula:
F = m * v² / r
Where:
F = centripetal force (tension in the string)
m = mass of the block
v = speed of the block
r = radius
Substituting in the given values:
F = (0.300 kg) * (76.5 cm/second)² / (54.0 cm)
Converting the units:
1 kg = 1000 g
1 m = 100 cm
F = (0.300 kg) * ((76.5 cm/second) / (100 cm/m))² / (54.0 cm)
F = (0.300 kg) * (0.765 m/second)² / (54.0 cm)
F = (0.300 kg) * (0.586225 m²/second²) / (54.0 cm)
F = 0.008544 m²/(kg*second²) / (54.0 cm)
Converting cm to meters:
1 m = 100 cm
F = 0.008544 m²/(kg*second²) / (0.54 m)
F = 0.0158 N
Therefore, the tension in the string is 0.0158 N.