A mass m1 = 5.8 kg rests on a frictionless table. It is connected by a massless and frictionless pulley to a second mass m2 = 2.3 kg that hangs freely.
1)What is the magnitude of the acceleration of block 1?
2)What is the tension in the string?
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Now the table is tilted at an angle of θ = 80° with respect to the vertical. Find the magnitude of the new acceleration of block 1.
4)At what “critical” angle will the blocks NOT accelerate at all?
5)Now the angle is decreased past the “critical” angle so the system accelerates in the opposite direction. If θ = 35° find the magnitude of the acceleration.
6)Compare the tension in the string in each of the above cases on the incline
To solve these problems, we can use Newton's laws of motion and the concepts of force, mass, and acceleration. I will guide you through the steps to find the answers.
1) To find the magnitude of the acceleration of block 1 in the given scenario, we need to consider that the masses are connected by a pulley. Since the table is frictionless, the tension in the string will be the same on both sides.
We can start by considering the forces acting on block 1. The only force acting on it is the tension in the string, directed upwards. According to Newton's second law (F = ma), the net force on block 1 is equal to mass times acceleration.
F_net = m1 * a
Since the tension in the string is the only force acting on block 1, we can substitute it into the equation:
Tension = m1 * a
Now, let's move on to the second question.
2) To find the tension in the string, we need to consider the forces acting on block 2. In this case, the only force acting on block 2 is its weight, which is equal to its mass (m2) multiplied by the acceleration due to gravity (g).
Weight of block 2 = m2 * g
Since the tension in the string is the force responsible for the acceleration of both blocks, we can equate the force due to the weight of block 2 to the tension:
Tension = m2 * g
Now, let's move on to the next scenario.
3) When the table is tilted at an angle of θ = 80° with respect to the vertical, the forces acting on the blocks will change. Gravity will still pull block 2 downwards, but it will now have a component that is parallel to the incline.
The magnitude of the new acceleration of block 1 can be found by considering the forces on block 1. The force due to gravity acting on block 1 can be broken down into two components: one parallel to the incline and one perpendicular to it.
The component of the weight acting parallel to the incline is given by:
Weight_parallel = m1 * g * sin(θ)
Since the table is frictionless, the net force parallel to the incline is the difference between the force due to gravity and the tension in the string:
F_parallel = Weight_parallel - Tension
We can set up the equation F_parallel = m1 * a and solve for the acceleration (a).
4) The "critical" angle refers to the angle at which the blocks stop accelerating and reach a state of equilibrium. In this case, the critical angle refers to the angle at which the force due to gravity on block 1 is equal to the force due to gravity on block 2.
At this critical angle, the net force on block 1 is zero, so the block will not accelerate. We can set up the equation:
m1 * g * sin(θ_c) = m2 * g
Solve for the critical angle (θ_c).
5) When the angle is decreased past the critical angle and the system accelerates in the opposite direction, we can use similar steps as in question 3.
Set up the equation F_parallel = m1 * a, taking into account the forces acting parallel to the incline, and solve for the acceleration (a) using the given angle (θ = 35°).
6) To compare the tension in the string in each of the above cases, you can use the equations mentioned earlier for finding the tension in the string. Substitute the respective masses and angles to calculate the tension in each scenario.