A 8.90-kg hanging object is connected by a light, inextensible cord over a light, frictionless pulley to a 5.00-kg block that is sliding on a flat table. Taking the coefficient of kinetic friction as 0.150, find the tension in the string.
First, we draw the free body diagrams and set up the equations for the problem. Let's denote the tension in the string T, the acceleration of the system a, the force of gravity on the hanging object FG, the force of kinetic friction Fk, and the weight of the block on the table W.
- For the 5.00-kg block on the table:
- The force pulling it horizontally is T.
- The force of kinetic friction Fk is acting in the opposite direction of the movement.
- The weight is W = m * g, where m = 5.00 kg and g = 9.81 m/s²
- The normal force Fn acts vertically upward and is equal to the weight W for the system to be in equilibrium in the vertical direction.
The equation for the horizontal direction can be written as:
m1 * a = T - Fk (1)
- For the 8.90-kg hanging object:
- The force of gravity FG = m * g, where m = 8.90 kg and g = 9.81 m/s²
- The tension T acts upward.
The equation for the vertical direction can be written as:
m2 * a = FG - T (2)
Now, let's calculate the force of gravity FG and the force of kinetic friction Fk.
FG = m2 * g = 8.90 kg * 9.81 m/s² = 87.309 N.
Fk = µ * Fn = µ * W = µ * m1 * g = 0.150 * 5.00 kg * 9.81 m/s² = 7.3575 N.
Next, we need to find the acceleration a of the system to solve for T.
We can eliminate T by summing equations (1) and (2):
m1 * a + m2 * a = FG - Fk
a = (FG - Fk) / (m1 + m2) = (87.309 N - 7.3575 N) / (5.00 kg + 8.90 kg) = 79.9515 N / 13.90 kg = 5.7509 m/s².
Finally, we can use this to find the tension T, either by plugging into equation (1) or (2). Let's use equation (1):
T = m1 * a + Fk = 5.00 kg * 5.7509 m/s² + 7.3575 N = 28.7545 N + 7.3575 N = 36.112 N.
The tension in the string is 36.112 N.
To find the tension in the string, we need to consider the forces acting on the system.
First, let's break down the forces on the hanging object:
1. Gravitational force (weight): This force can be calculated using the formula F = mg, where m is the mass of the object and g is the acceleration due to gravity (9.8 m/s²). In this case, the weight of the object is F1 = 8.90 kg * 9.8 m/s².
Now, let's analyze the forces on the sliding block:
1. Normal force: This force arises due to the contact between the block and the table. It is equal in magnitude and opposite in direction to the gravitational force acting on the block. Here, F2 = 5.00 kg * 9.8 m/s².
2. Friction force: The friction force opposes the motion of the block. It can be calculated using the formula F_friction = μ * F_normal, where μ is the coefficient of kinetic friction. Here, F_friction = 0.150 * F_normal.
Now, we can calculate the tension in the string. The tension is the same on both sides of the pulley.
The equation to determine the tension is: Tension = F1 - F_friction.
Let's substitute the values:
Tension = (8.90 kg * 9.8 m/s²) - (0.150 * (5.00 kg * 9.8 m/s²)).
Once you solve this equation, you'll find the tension in the string.
To find the tension in the string, we need to analyze the forces acting on the system. Let's break it down into steps:
Step 1: Determine the weight of the hanging object and the block on the table.
The weight of an object is given by the formula: weight = mass x acceleration due to gravity (g).
Weight of hanging object = mass of hanging object x g
= 8.90 kg x 9.8 m/s² (acceleration due to gravity)
= 87.02 N
Weight of block on the table = mass of block x g
= 5.00 kg x 9.8 m/s²
= 49.00 N
Step 2: Calculate the force of friction acting on the block.
The force of friction is given by the formula: force of friction = coefficient of friction x normal force.
The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, it is equal to the weight of the block on the table.
Normal force = weight of block on the table
= 49.00 N
Force of friction = coefficient of friction x normal force
= 0.150 x 49.00 N
= 7.35 N
Step 3: Determine the net force acting on the system.
The net force is the sum of all the forces acting on the system.
Net force = (force of hanging object) - (force of friction)
= (weight of hanging object) - (force of friction)
= 87.02 N - 7.35 N
= 79.67 N
Step 4: Determine the acceleration of the system.
The acceleration of the system can be calculated using Newton's second law: net force = mass x acceleration.
Acceleration = net force / (mass of hanging object + mass of block)
= 79.67 N / (8.90 kg + 5.00 kg)
= 5.46 m/s²
Step 5: Find the tension in the string.
The tension in the string is the force that allows the system to accelerate. In this case, it is the force exerted by the hanging object.
Tension in the string = weight of hanging object - force of gravity on the hanging object
= (mass of hanging object x g) - (mass of hanging object x acceleration)
= (8.90 kg x 9.8 m/s²) - (8.90 kg x 5.46 m/s²)
= 79.62 N
Therefore, the tension in the string is 79.62 N.