A block of mass M resting on a 21.5° slope is shown. The block has coefficients of friction μs=0.605 and μk=0.344 with the surface. It is connected via a massless string over a massless, frictionless pulley to a hanging block of mass 2.57 kg. What is the minimum mass M1 that will stick and not slip?
I am uncertain what M1 is.
To find the minimum mass M1 that will stick and not slip, we need to determine the maximum force of friction that the surface can provide.
Let's break down the problem step-by-step:
1. Calculate the weight of the hanging block:
Weight = mass × acceleration due to gravity
= 2.57 kg × 9.8 m/s^2
= 25.1266 N
2. Determine the force of friction at the slope:
Since the block does not accelerate, the force of friction must equal the component of the weight of the hanging block perpendicular to the slope.
Force of friction = Weight × perpendicular component of the weight
= Weight × sin(21.5°)
= 25.1266 N × sin(21.5°)
= 8.8284 N
3. Calculate the maximum force of friction:
The maximum force of friction can be determined using the coefficient of static friction (μs):
Maximum force of friction = μs × normal force
= μs × (mass of the block + mass of the hanging block) × acceleration due to gravity
= 0.605 × (M + 2.57 kg) × 9.8 m/s^2
4. Equate the maximum force of friction to the force of friction at the slope:
0.605 × (M + 2.57 kg) × 9.8 m/s^2 = 8.8284 N
5. Solve for M:
M + 2.57 kg = 8.8284 N / (0.605 × 9.8 m/s^2)
M = (8.8284 N / (0.605 × 9.8 m/s^2)) - 2.57 kg
Calculating the value, we find:
M ≈ 1.3008 kg
Therefore, the minimum mass M1 that will stick and not slip is approximately 1.3008 kg.