A crate of mass 54 kg is attached to one end of a string, and the other end of the string runs over a pulley and is held by a person as in the figure below. If the person pulls with a force of 93 N, what is the crate's acceleration? (Assume that upward is the positive direction. Indicate the direction with the sign of your answer.)
To find the crate's acceleration, we'll use Newton's second law of motion which states that the force applied to an object equals its mass times its acceleration:
F = m * a
where F is the force applied (93 N) and m is the mass of the crate (54 kg).
First, let's determine the direction of the acceleration. Since the person is pulling the crate upwards, we'll assume the positive direction is upwards.
Now let's plug in the values into the equation:
93 N = 54 kg * a
To solve for acceleration (a), we'll divide both sides by the mass (54 kg):
a = 93 N / 54 kg
a ≈ 1.72 m/s²
Therefore, the crate's acceleration is approximately 1.72 m/s² in the upward direction.
To find the crate's acceleration, we can start by applying Newton's second law of motion, which states that the acceleration of an object is directly proportional to the net force applied to it and inversely proportional to its mass.
Step 1: Identify the forces acting on the crate.
In this scenario, the crate is being pulled upwards by the person with a force of 93 N. We also have the force of gravity acting downwards, which can be calculated using the equation F = m * g, where m is the mass of the crate and g is the acceleration due to gravity, approximately 9.8 m/s^2.
Force of gravity = 54 kg * 9.8 m/s^2
Force of gravity = 529.2 N (directed downwards)
Step 2: Calculate the net force acting on the crate.
The net force is the vector sum of all the forces acting on the crate. In this case, since the person is pulling upwards and the force of gravity is acting downwards, we can subtract the force of gravity from the force applied by the person to get the net force.
Net force = Force applied - Force of gravity
Net force = 93 N - 529.2 N
Net force = -436.2 N (negative sign indicates that the net force is downward)
Step 3: Calculate the acceleration of the crate.
Using Newton's second law (F = m * a), we can rearrange the equation to solve for acceleration:
acceleration = net force / mass
a = (-436.2 N) / (54 kg)
a = -8.08 m/s^2
So, the crate's acceleration is -8.08 m/s^2, with the negative sign indicating that it is directed downward.