3.92 kg block located on a horizontal floor is pulled by a cord that exerts a force F = 11.5 N at an angle theta = 15.5o above the horizontal, as shown in the Figure. The coefficient of kinetic friction between the block and the floor is 0.09. What is the speed of the block 5.1s after it starts moving?
Well, I will sketch out this one but then you try the next one.
force down on floor = 3.92 g - 11.5 sin 15.5
so
friction force = 0.09(3.92 g - 11.5 sin 15.5)
call it ff
then
11.5 cos 15.5 - ff = 3.92 a
calculate a
initial speed = 0
so
v = a t = a * 5.1 meters/second
To find the speed of the block 5.1s after it starts moving, we can use the concept of Newton's second law of motion and the equations of motion.
Here are the steps to solve the problem:
1. First, let's analyze the forces acting on the block. These forces include:
- The force pulling the block horizontally from the cord, which can be split into horizontal and vertical components.
- The force of kinetic friction between the block and the floor.
- The normal force exerted by the floor on the block, which is equal in magnitude but opposite in direction to the vertical component of the force pulling the block.
2. Since the block is moving horizontally, the force pulling the block in the horizontal direction (F_horizontal) can be determined using the following equation:
F_horizontal = F * cos(theta)
where F is the magnitude of the force (11.5 N) and theta is the angle (15.5 degrees).
F_horizontal = 11.5 N * cos(15.5 degrees)
F_horizontal ≈ 11.03 N
3. The force of kinetic friction (F_friction) can be calculated using the equation:
F_friction = μ * N
where μ is the coefficient of kinetic friction (0.09) and N is the normal force.
To find the normal force, we can use the vertical component of the force pulling the block:
N = F * sin(theta)
N = 11.5 N * sin(15.5 degrees)
N ≈ 3.02 N
Now, we can calculate the force of kinetic friction:
F_friction = 0.09 * 3.02 N
F_friction ≈ 0.27 N
4. By applying Newton's second law of motion in the horizontal direction, we can write the equation:
F_net = ma
where F_net is the net force on the block, m is the mass of the block (3.92 kg), and a is the acceleration of the block.
The net force can be calculated by subtracting the force of kinetic friction from the horizontal pulling force:
F_net = F_horizontal - F_friction
F_net ≈ 11.03 N - 0.27 N
F_net ≈ 10.76 N
Now, we can calculate the acceleration:
a = F_net / m
a ≈ 10.76 N / 3.92 kg
a ≈ 2.74 m/s^2
5. Using the equations of motion, we can find the final velocity (v) of the block after 5.1 seconds, given the initial velocity (u = 0 m/s) and acceleration (a = 2.74 m/s^2):
v = u + a * t
v = 0 m/s + 2.74 m/s^2 * 5.1 s
v ≈ 13.974 m/s
Therefore, the speed of the block 5.1 seconds after it starts moving is approximately 13.974 m/s.