A 220g block on a 54.0cm -long string swings in a circle on a horizontal, frictionless table at 70.0 rpm.

A) What is the speed of the block? m/s
B) What is the tension in the string? N

To solve this problem, we can use the formulas for the speed of an object moving in a circle and the centripetal force.

A) To find the speed of the block, we can use the formula:

speed = 2πr / T

where,
speed = velocity of the block (m/s)
π ≈ 3.14
r = radius of the circle (m)
T = time taken for one complete revolution (s)

Given that the block is moving on a circular path of radius 54.0 cm or 0.54 m and completes one revolution in 70.0 rpm which is equal to 1.17 Hz (since 1 Hz = 60 rpm):

speed = 2π(0.54) / 1.17
speed = 3.14 * 0.54 / 1.17
speed ≈ 1.44 m/s

Therefore, the speed of the block is approximately 1.44 m/s.

B) To find the tension in the string, we can use the formula for the centripetal force:

F = m * v^2 / r

where,
F = centripetal force (N)
m = mass of the block (kg)
v = velocity of the block (m/s)
r = radius of the circle (m)

Given that the mass of the block is 220g, or 0.22 kg, and the velocity of the block is approximately 1.44 m/s, we can substitute these values into the formula:

F = (0.22 kg) * (1.44 m/s)^2 / 0.54 m
F = (0.22) * (2.0736) / 0.54
F = 0.451392 / 0.54
F ≈ 0.834 N

Therefore, the tension in the string is approximately 0.834 N.

To find the speed of the block, we need to convert the given rotation rate in revolutions per minute (rpm) to radians per second (rad/s). Then we can use the formula for linear velocity to calculate the speed.

A) First, we convert 70.0 rpm to radians per second (rad/s):
1 revolution = 2π radians
1 minute = 60 seconds

70.0 rpm = 70.0 * 2π radians/minute
= 70.0 * 2π / 60 radians/second
≈ 7.33 radians/second

Now, we can calculate the speed of the block using the formula for linear velocity:
v = rω

In this case, the radius (r) is given as 54.0 cm (convert to meters by dividing by 100) and the angular velocity (ω) is 7.33 radians/second.

v = (0.54 m) * (7.33 rad/s)
≈ 3.96 m/s

Therefore, the speed of the block is approximately 3.96 m/s.

B) To find the tension in the string, we can use the centripetal force formula:

F = mv^2 / r

where F is the tension in the string, m is the mass of the block, v is the speed of the block, and r is the radius of the circular path.

In this case, the mass (m) is given as 220 g (convert to kilograms by dividing by 1000), the speed (v) is 3.96 m/s, and the radius (r) is 54.0 cm (convert to meters by dividing by 100).

F = (0.22 kg) * (3.96 m/s)^2 / (0.54 m)
≈ 6.62 N

Therefore, the tension in the string is approximately 6.62 N.