In the figure below, block 1 of mass
m1 = 3.0 kg
and block 2 of mass
m2 = 1.0 kg
are connected by a string of negligible mass. Block 2 is pushed by force F of magnitude 30 N and angle
θ = 20°.
The coefficient of kinetic friction between each block and the horizontal surface is 0.25. What is the tension in the string?
To find the tension in the string, we need to analyze the forces acting on the system.
1. Start by resolving the force applied at an angle into its horizontal and vertical components.
- The horizontal component of the force is F*cos(θ), where F is the magnitude of the force and θ is the angle.
- The vertical component of the force is F*sin(θ).
2. Next, consider the forces acting on each block individually:
- For block 1:
- The weight of block 1 is m1 * g, where g is the acceleration due to gravity (approximately 9.8 m/s²).
- The tension in the string is T, acting in the opposite direction.
- The frictional force opposing the motion of block 1 is μ1 * N1, where μ1 is the coefficient of kinetic friction and N1 is the normal force.
- For block 2:
- The weight of block 2 is m2 * g.
- The tension in the string is T, acting in the same direction.
- The frictional force opposing the motion of block 2 is μ2 * N2, where μ2 is the coefficient of kinetic friction and N2 is the normal force.
3. Use Newton's second law, F = ma, for each block:
- For block 1: F_net1 = m1 * a1
- The net force acting in the horizontal direction is T - μ1 * N1 = m1 * a1.
- The net force acting in the vertical direction is N1 - m1 * g = 0 (since block 1 is not accelerating vertically).
- For block 2: F_net2 = m2 * a2
- The net force acting in the horizontal direction is T - F*cos(θ) - μ2 * N2 = m2 * a2.
- The net force acting in the vertical direction is N2 - m2 * g = -m2 * g (since block 2 is not accelerating vertically).
4. Determine the normal forces for each block:
- The normal force, N1, can be calculated by N1 = m1 * g (since block 1 is not accelerating vertically).
- The normal force, N2, can be calculated by N2 = m2 * g + F*sin(θ) (since block 2 is accelerating vertically).
5. Substitute the normal forces back into the net force equations and solve for the tension, T.
Once you have the values for the normal forces, you can use them to find the tension in the string by solving the net force equations.
To find the tension in the string, we need to analyze the forces acting on the blocks. Let's break it down step by step:
Step 1: Calculate the gravitational force on each block.
Using the equation F_gravity = m * g, where m is the mass and g is the acceleration due to gravity (approximately 9.8 m/s^2), we can calculate the gravitational force on each block.
For block 1:
F_gravity_1 = m1 * g
= 3.0 kg * 9.8 m/s^2
= 29.4 N
For block 2:
F_gravity_2 = m2 * g
= 1.0 kg * 9.8 m/s^2
= 9.8 N
Step 2: Calculate the normal force on each block.
The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, the normal force will be equal to the gravitational force for each block since they are not accelerating in the vertical direction.
For block 1: F_normal_1 = F_gravity_1 = 29.4 N
For block 2: F_normal_2 = F_gravity_2 = 9.8 N
Step 3: Calculate the frictional force on each block.
The frictional force can be calculated using the equation F_friction = μ * F_normal, where μ is the coefficient of friction and F_normal is the normal force.
For block 1: F_friction_1 = μ * F_normal_1 = 0.25 * 29.4 N = 7.35 N
For block 2: F_friction_2 = μ * F_normal_2 = 0.25 * 9.8 N = 2.45 N
Step 4: Calculate the force in the x-direction for block 2.
The force provided by the applied force F can be divided into the x and y-components. The x-component can be calculated as F_x = F * cos(θ).
F_x = 30 N * cos(20°)
= 30 N * 0.9397
≈ 28.191 N
Step 5: Calculate the net force in the x-direction for block 2.
The net force in the x-direction can be calculated as F_net_x = F_x - F_friction_2.
F_net_x = 28.191 N - 2.45 N
= 25.741 N
Step 6: Calculate the acceleration of block 2.
Using Newton's second law, F_net = m * a, we can solve for the acceleration.
F_net_x = m2 * a
25.741 N = 1.0 kg * a
a ≈ 25.741 m/s^2
Step 7: Calculate the tension in the string.
Since block 1 and block 2 are connected by a string, they will experience the same tension force. Therefore, the tension in the string will be equal to the net force in the x-direction for block 1.
Tension = F_net_x for block 1
= F_friction_1 + F_net_x
= 7.35 N + 25.741 N
= 33.091 N
Therefore, the tension in the string is approximately 33.091 N.