A 27.9 kg block (m1) is on a horizontal surface, connected to a 5.3 kg block (m2) by a massless string. The pulley is massless and frictionless. A force of 239.3 N acts on m1 at an angle of 28.9°. The coefficient of kinetic friction between m1 and the surface is 0.181. Determine the upward acceleration of m2.
To determine the upward acceleration of m2, we need to follow a step-by-step approach. Here's how you can solve this problem:
1. Draw a free-body diagram for each block:
- For block m1:
- The force acting on it is the tension in the string, T.
- The weight of m1, which is equal to m1 * g, where g is the acceleration due to gravity.
- The frictional force acting on m1, which is equal to the coefficient of kinetic friction (μk) multiplied by the normal force (N), where N = m1 * g.
- For block m2:
- The force acting on it is the tension in the string, T.
- The weight of m2, which is equal to m2 * g.
2. Write down the equations of motion for each block:
- For m1:
- ΣF = T - μk * N = m1 * a1, where a1 is the acceleration of m1.
- For m2:
- ΣF = T - m2 * g = m2 * a2, where a2 is the acceleration of m2.
3. Use the given information to find the values needed for the equations:
- m1 = 27.9 kg
- m2 = 5.3 kg
- Force acting on m1 = 239.3 N
- Angle of the force = 28.9°
- Coefficient of kinetic friction (μk) = 0.181
- Acceleration due to gravity (g) ≈ 9.8 m/s^2
4. Resolve the force acting on m1 into its components:
- The horizontal component, F_h = Force * cos(angle)
- The vertical component, F_v = Force * sin(angle)
5. Calculate the normal force for m1:
- N = m1 * g
6. Substitute the values into the equations of motion and solve for the unknowns:
First, let's solve for the tension in the string (T):
- From the equation for m2: T = m2 * g + m2 * a2
Next, let's solve for the acceleration of m1 (a1):
- From the equation for m1: T - μk * N = m1 * a1
Finally, we can substitute the value of T into the equation for m2 and solve for the acceleration of m2 (a2):
- T = m2 * g + m2 * a2
Replace T with the value you found in the previous step and solve for a2.
7. Calculate the upward acceleration of m2.
- Substitute the known values into the equation and solve for a2.
By following these steps, you should be able to determine the upward acceleration of m2.
To determine the upward acceleration of m2, we need to consider the forces acting on both blocks.
For m1:
- The force applied at an angle of 28.9° can be resolved into two components:
- The vertical component: F_vert = 239.3 N * sin(28.9°)
- The horizontal component: F_horiz = 239.3 N * cos(28.9°)
- The gravitational force acting downward: F_gravity_m1 = m1 * g, where g is the acceleration due to gravity (9.8 m/s^2).
- The force of kinetic friction opposing the motion: F_friction = μ * F_normal, where μ is the coefficient of kinetic friction and F_normal is the normal force. Since the block is on a horizontal surface, F_normal = m1 * g.
- The net force acting on m1 in the horizontal direction: F_net_m1 = F_horiz - F_friction
For m2:
- The gravitational force acting downward: F_gravity_m2 = m2 * g
Since the blocks are connected by a massless string, the tension in the string is the same for both blocks:
- Tension in the string: T = m2 * a (where a is the common acceleration of both blocks)
Now, we can equate the net force on m1 with the tension in the string:
F_net_m1 = T
Let's calculate the values step by step.
Step 1: Calculate the forces acting on m1:
F_vert = 239.3 N * sin(28.9°)
F_horiz = 239.3 N * cos(28.9°)
F_gravity_m1 = m1 * g
F_normal = m1 * g
F_friction = μ * F_normal
Step 2: Calculate the net force acting on m1:
F_net_m1 = F_horiz - F_friction
Step 3: Equate the net force on m1 with the tension in the string:
F_net_m1 = T
Step 4: Calculate the tension in the string:
T = m2 * a
Step 5: Equate the tension in the string with the gravitational force on m2:
T = F_gravity_m2
Step 6: Solve for the acceleration, a:
m2 * a = F_gravity_m2
Finally, we can substitute the values and calculate the upward acceleration of m2.