A block of mass m1 = 41.3 kg on a horizontal surface is connected to a mass m2 = 13.2 kg that hangs vertically as shown in the figure below. The two blocks are connected by a string of negligible mass passing over a frictionless pulley. Assume that the horizontal surface is smooth
(a) What is the acceleration of the hanging mass? (Enter the magnitude only.)
(b) Determine the tension in the cord. (Enter the magnitude only.)
Why did the block go to therapy?
Because it had tension issues!
Now, let's solve your physics problem.
(a) To find the acceleration, we can use Newton's second law. The net force acting on the system is given by the weight of the hanging mass (m2g) minus the tension in the cord. So we have:
m2g - T = m2a, where g is the acceleration due to gravity.
Simplifying, we get:
T = m2g - m2a
Now, since the blocks are connected and move together, the acceleration of the hanging mass (m2) will be the same as the acceleration of the block (m1). Let's call it "a".
So we can rewrite the equation as:
T = m2g - m2a = m2g - m1a
Now, we know the values for m1 and m2, so we can substitute them in:
T = (13.2 kg) * (9.8 m/s^2) - (41.3 kg) * a
Solving for a, we get:
a = (13.2 kg * 9.8 m/s^2) / (13.2 kg + 41.3 kg)
Calculating this, we find:
a ≈ 2.48 m/s^2
So, the acceleration of the hanging mass is approximately 2.48 m/s^2.
(b) To find the tension in the cord, we can use the following equation:
T = m2g - m1a
Substituting the known values, we have:
T = (13.2 kg) * (9.8 m/s^2) - (41.3 kg) * (2.48 m/s^2)
Calculating this, we find:
T ≈ 98.2 N
So, the tension in the cord is approximately 98.2 N.
I hope that made you smile while solving your physics problem!
To find the acceleration of the hanging mass and the tension in the cord, we can use Newton's second law of motion and some basic principles of physics.
First, let's analyze the forces acting on the system. For the hanging mass, there are two main forces: its weight (mg) pulling it downwards and the tension in the cord pulling it upwards. For the block on the horizontal surface, there is only one force, which is the tension in the cord acting to the right.
Now, let's calculate the acceleration of the hanging mass (m₂). Since the pulley is frictionless and the horizontal surface is smooth, we can assume that the tension in the cord is the same on both sides. Therefore, the force pulling the hanging mass upwards is equal to the force pulling the block on the horizontal surface to the right.
Using Newton's second law, we can write the equation of motion for each object:
For the hanging mass (m₂):
m₂ * a = T - m₂ * g
For the block on the horizontal surface (m₁):
m₁ * a = T
Where:
m₂ = mass of the hanging mass (13.2 kg)
m₁ = mass of the block on the horizontal surface (41.3 kg)
a = acceleration (unknown)
T = tension in the cord (unknown)
g = acceleration due to gravity (9.8 m/s²)
Since both equations involve the same tension T, we can equate them and solve for the acceleration (a):
m₂ * a = T - m₂ * g
m₁ * a = T
T - m₂ * g = m₁ * a
Plugging in the given values:
T - (13.2 kg) * (9.8 m/s²) = (41.3 kg) * a
Now, we need to solve for the acceleration (a). Rearranging the equation:
T = (41.3 kg) * a + (13.2 kg) * (9.8 m/s²)
Now we can substitute the values into the equation and solve for acceleration (a):
T = (41.3 kg) * a + (13.2 kg) * (9.8 m/s²)
T = 41.3a + 129.36
Now we can solve for the tension (T) by substituting the value of acceleration (a) in the first equation:
T = (41.3 kg) * a + (13.2 kg) * (9.8 m/s²)
T = (41.3 kg) * (T - (13.2 kg) * (9.8 m/s²)) + (13.2 kg) * (9.8 m/s²)
T = 41.3T - 541.2 + 129.36
T - 41.3T = -541.2 + 129.36
-40.3T = -411.84
T = (-411.84) / (-40.3)
Finally, we can calculate the magnitude of the tension (T) and the acceleration (a) by plugging in the numbers:
(a) The acceleration of the hanging mass is |a| = |-411.84 / -40.3| = 10.22 m/s² (rounded to two decimal places).
(b) The tension in the cord is |T| = |-411.84 / -40.3| = 10.22 N (rounded to two decimal places).