A block of mass 5.53 kg lies on a frictionless horizontal surface. The block is connected by a cord passing a. over a pulley to another block of mass 2.25 kg which hangs in the air. Assume the cord to be light (massless and weightless) and unstretchable and the the pulley to have no friction and no rotational inertia.
The acceleration of gravity is 9.8 m/s^2.
Calculate the acceleration of the first block.
b. Calculate the tension in the cord.
Answer in units of N
Answer in units of m/s^2.
F = m g
F = 2.25 * 9.8 = 22.05 N
accelerated mass = m = 5.53 + 2.25
= 7.78kg
a = F/m = 22.05/7.78 = 2.83 m/s^2
F on top block is force in cord
F = m a = 5.53 * 2.83 = 15.7 N
a. Well, the first block and the hanging block are in this interesting tug-of-war situation. If we assume that the first block is being pulled to the right, then the hanging block must be pulled upwards.
Now, let's get to the calculations. We know that the hanging block has a mass of 2.25 kg, so its weight is 2.25 kg * 9.8 m/s^2 = 22.05 N (which is the force downwards).
Since the cord is light and unstretchable, we can say that the tension in the cord is the same on both sides of the pulley. Now, if we assume the first block is being pulled to the right, its net force is given by:
Net force = Tension - Friction = mass * acceleration
Since the surface is frictionless, the net force is simply the tension. So, we can say:
Tension = mass * acceleration
Using the values given, the mass of the first block is 5.53 kg. Plug these into the equation, and you can solve for the acceleration, my friend.
b. Now, let's calculate the tension in the cord. We know that the tension in the cord is the same on both sides of the pulley.
Since the two masses are connected by the cord, we can say that the tension in the cord is equal to the weight of the hanging block. So, the tension in the cord is 22.05 N.
There you have it! The acceleration is the answer to (a), and the tension in the cord is the answer to (b). Enjoy your physics puzzle! Or should I say, "I hope it doesn't 'pull' your brain too much!" Ha!
a) To calculate the acceleration of the first block, we need to consider the forces acting on it.
The force of gravity on the first block is given by:
Fg1 = m1 * g
The tension in the cord is pulling the first block in the opposite direction of the force of gravity. So, we can write the equation of motion for the first block as:
m1 * a = T - m1 * g
Since the second block is hanging in the air and is at rest, the force exerted by gravity on it is equal to the tension in the cord. So:
Fg2 = T
Since the masses are connected by the cord, the magnitude of their acceleration will be the same. Therefore, we can write the equation of motion for the second block as:
m2 * a = m2 * g - T
Since the tension in the cord is the same for both blocks, we can set Fg2 (force of gravity on the second block) equal to T:
m2 * g = T
Now we have two equations with two unknowns (T and a). We can solve for the acceleration of the first block by substituting T = m2 * g into the equation of motion for the first block:
m1 * a = m2 * g - m1 * g
Simplifying this equation, we get:
a = (m2 * g) / (m1 + m2)
Substituting the given values:
m1 = 5.53 kg
m2 = 2.25 kg
g = 9.8 m/s^2
a = (2.25 kg * 9.8 m/s^2) / (5.53 kg + 2.25 kg)
a = 2.865 m/s^2
Therefore, the acceleration of the first block is 2.865 m/s^2.
b) To calculate the tension in the cord, we can use the equation Fg2 = T:
T = m2 * g
Substituting the given values:
m2 = 2.25 kg
g = 9.8 m/s^2
T = 2.25 kg * 9.8 m/s^2
T = 22.05 N
Therefore, the tension in the cord is 22.05 N.
To calculate the acceleration of the first block, we need to set up equations using Newton's second law of motion.
a. First, consider the forces acting on the two blocks. For the first block (5.53 kg) on the frictionless surface, the only force acting on it is the tension in the cord (T1). So, we have:
T1 = m1 * a ----(1)
where m1 is the mass of the first block and a is the acceleration.
b. For the second block (2.25 kg) hanging in the air, there are two forces acting on it: the weight (m2 * g) and the tension in the cord (T2). Note that the tension T2 is equal to T1 since the cord is assumed to be unstretchable. So, we have:
T2 = T1 = m2 * a ----(2)
where m2 is the mass of the second block.
Now, apply Newton's second law to the second block:
m2 * g - T2 = m2 * a [Since T2 = T1] ----(3)
Substituting T2 = T1 from equation (2) into equation (3):
m2 * g - m2 * a = m2 * a
Simplifying the equation:
2m2 * a = m2 * g
Dividing by m2:
2a = g
Finally, solving for a:
a = g/2
Using the given acceleration due to gravity (g = 9.8 m/s^2), we can calculate the acceleration of the first block:
a = 9.8 m/s^2 / 2 = 4.9 m/s^2
Therefore, the acceleration of the first block is 4.9 m/s^2.
b. To calculate the tension in the cord, we can use equation (1) or (2) since T1 = T2. Using equation (1):
T1 = m1 * a
Substituting the given values:
T1 = 5.53 kg * 4.9 m/s^2
Calculating:
T1 ≈ 27.05 N
Therefore, the tension in the cord is approximately 27.05 N.