A 220 g block on a 56.0 cm -long string swings in a circle on a horizontal, frictionless table at 50.0 rpm
a) find speed of the block
b) whats the tension in the string
thanks!
a) To find the speed of the block, we first have to convert the rpm to rad/s. We know that 1 revolution is equal to 2π radians, so the conversion factor is (2π rad/rev). We can then use the formula:
ω = 50.0 (rev/min) * (2π rad/rev) * (1 min/60 s)
Computing this as:
ω = 50.0 * (2 * 3.14159) * (1/60) = 5.236/s
Now that we have the angular velocity, we can find the linear velocity of the block by multiplying it by the radius of the circle:
v = r * ω
The radius is half of the string's length, which is 56.0 cm / 2 = 28.0 cm = 0.28 m. So we have:
v = 0.28 m * 5.236 s^-1 = 1.466 m/s
The speed of the block is 1.466 m/s.
b) To find the tension in the string, we will use the centripetal force equation:
Fc = m * v² / r
Where m is the mass of the block (220 g = 0.22 kg), v is the speed of the block (1.466 m/s), and r is the radius (0.28 m). Plugging these values into the equation, we get:
Fc = 0.22 kg * (1.466 m/s)² / 0.28 m = 1.497 N
The tension in the string is 1.497 N.
Sure! Let's break it down step-by-step.
Step 1: Convert the rotational speed from rpm to rad/s.
The formula to convert from rpm to rad/s is:
rad/s = (rpm) * (2π/60)
Given that the block swings at 50.0 rpm, we can calculate the rotational speed as follows:
rotational speed = (50.0 rpm) * (2π/60) = 5.24 rad/s
Step 2: Calculate the speed of the block.
The speed of an object moving in a circular path can be determined using the formula:
speed = (radius) * (rotational speed)
Here, the radius is the length of the string, which is 56.0 cm or 0.56 m.
Substituting the values into the formula, we get:
speed = (0.56 m) * (5.24 rad/s) ≈ 2.93 m/s
a) The speed of the block is approximately 2.93 m/s.
Step 3: Calculate the tension in the string.
The tension in the string can be found using the centripetal force formula:
tension = (mass) * (speed²) / (radius)
Given that the mass of the block is 220 g or 0.220 kg, and the radius of the circular path is 0.56 m, we can substitute these values into the formula:
tension = (0.220 kg) * (2.93 m/s)² / (0.56 m) ≈ 2.43 N
b) The tension in the string is approximately 2.43 N.
I hope this helps! Let me know if you have any further questions.
To find the speed of the block, we can use the formula for the circumference of a circle and the given angular velocity.
a) The formula for the circumference of a circle is C = 2πr, where C represents the circumference and r represents the radius.
Given that the block swings in a circle with a length of 56.0 cm, we can calculate the radius by dividing the length by 2π: r = 56.0 cm / (2π) ≈ 8.91 cm.
The angular velocity is given as 50.0 rpm (revolutions per minute). To convert this to radians per second, we multiply by 2π/60: ω = (50.0 rpm) × (2π/60) ≈ 5.24 rad/s.
Now, we can calculate the speed of the block using the formula v = rω: v = (8.91 cm) × (5.24 rad/s) ≈ 46.7 cm/s.
Therefore, the speed of the block is approximately 46.7 cm/s.
b) To find the tension in the string, we can use the centripetal force formula, which states that the tension in the string is equal to the centripetal force required to keep the block moving in a circular path.
The centripetal force is given by the equation Fc = m * (v^2 / r), where Fc represents the centripetal force, m is the mass of the block, v is the speed of the block, and r is the radius.
Given that the mass of the block is 220 g, the speed is approximately 46.7 cm/s (which can be converted to m/s by dividing by 100), and r is 8.91 cm (which can be converted to m by dividing by 100), we have:
Fc = (220 g) * ((46.7 cm/s)^2 / (8.91 cm))
Fc = (0.220 kg) * ((0.467 m/s)^2 / (0.0891 m))
Calculating this expression gives us the centripetal force.
Finally, since the tension in the string is equal to the centripetal force, the tension in the string is the same value as the calculated centripetal force.