A 1.5 kg block is connected by a rope across a 50-cm-diameter, 2.0 kg, frictionless pulley. A constant 10 N tension is applied to the other end of the rope. Starting from rest, how long does it take the block to move 30 cm?
Is the block on a horizontal table, or hanging vertically?
hanging vertically
To find the time it takes for the block to move 30 cm, we need to use Newton's second law and the equations of motion.
First, we need to find the net force acting on the block. The tension in the rope, T, is equal to the force applied minus the force due to the weight of the block. The force due to the weight of the block, Fg, can be calculated using the formula Fg = m * g, where m is the mass of the block and g is the acceleration due to gravity (approximately 9.8 m/s²). In this case, the mass of the block is given as 1.5 kg.
Fg = m * g
Fg = 1.5 kg * 9.8 m/s²
Fg = 14.7 N
Now, we can calculate the tension in the rope:
T = 10 N - 14.7 N
T = -4.7 N
Since the tension is negative, it means that the block will move in the opposite direction of the applied force.
Next, we can use the equation of motion to find the acceleration of the block:
F = ma
-4.7 N = 1.5 kg * a
a = -4.7 N / 1.5 kg
a = -3.13 m/s²
Since the acceleration is negative, it means that the block will decelerate during the motion.
To find the time it takes for the block to move 30 cm, we can use the kinematic equation:
s = ut + 0.5at²
where:
s = displacement (30 cm = 0.3 m)
u = initial velocity (0 m/s, since the block starts from rest)
a = acceleration (-3.13 m/s²)
Plugging these values into the equation, we get:
0.3 m = 0 + 0.5 * (-3.13 m/s²) * t²
Simplifying the equation:
0.3 m = -1.57 m/s² * t²
t² = (0.3 m) / (-1.57 m/s²)
t² = -0.1911 s²
We discard the negative square root since time cannot be negative:
t = √(-0.1911) s
t = 0.437 s
Therefore, it takes approximately 0.437 seconds for the block to move 30 cm.