A slippery plastic block is resting in a ramp that makes an angle of 20 degrees with the horizontal. The coefficient of static friction between the block and ramp is 0.2. The coefficient of kinetic friction is 0.15.
A)will the block start sliding on its own
B)if you found that the block slid in part a, calculate its acceleration. If not, then calculate the minimum angle the ramp just have to start the block sliding
normal force= m g cos 20
so
max static friction force = 0.2 m g cos 20
component of weight down slope = m g sin 20
Net force down slope = m g (sin 20 - 0.2 cos 20)
= m g (0.342 - 0.188)
that is positive so it starts sliding
then we need to use the kinetic friction
Net force now = m g (sin 20 - 0.15 cos 20)
= m g ( 0.342 - 0.141) = m g (0.201)
so
m a = m g (0.201)
a = 0.201 * 9.81 on earth approximately = 1.97 m/s^2
To determine whether the block will start sliding on its own, we need to compare the force of gravity acting down the ramp with the maximum force of static friction that can be exerted by the block.
A) Calculate the force of gravity acting down the ramp:
F_gravity = m * g * sin(theta)
where m is the mass of the block, g is the acceleration due to gravity (approximately 9.8 m/s^2), and theta is the angle of the ramp.
B) Calculate the maximum force of static friction:
F_static = m * g * cos(theta) * mu_static
where mu_static is the coefficient of static friction between the block and the ramp.
If the force of gravity is greater than or equal to the maximum force of static friction (F_gravity ≥ F_static), the block will start sliding. If F_gravity < F_static, the block will not start sliding.
If the block starts sliding, we can calculate its acceleration using the kinetic friction force.
C) Calculate the force of kinetic friction:
F_kinetic = m * g * cos(theta) * mu_kinetic
where mu_kinetic is the coefficient of kinetic friction between the block and the ramp.
D) Calculate the acceleration:
a = (F_gravity - F_kinetic) / m
If the block does not start sliding in part A, we can calculate the minimum angle required to start the block sliding by setting the force of gravity equal to the maximum force of static friction:
F_gravity = F_static
Let's plug in the given values and calculate each step:
To determine whether the block will start sliding on its own, we need to compare the force of static friction to the maximum static friction force that can be exerted between the block and ramp. If the force of static friction is less than the maximum static friction force, the block will start sliding.
The maximum static friction force (F_max_static) can be calculated using the formula:
F_max_static = coefficient of static friction (μ_static) * normal force (N)
So, let's break down the problem step by step:
Step 1: Find the normal force (N)
The normal force (N) exerted on the block is equal to its weight (W), which can be calculated using the formula:
W = mass (m) * gravitational acceleration (g)
The gravitational acceleration (g) is approximately 9.8 m/s².
Step 2: Calculate the maximum static friction force (F_max_static)
Using the coefficient of static friction (μ_static) given as 0.2 and the normal force (N) calculated in step 1, we can determine the maximum static friction force:
F_max_static = 0.2 * N
Step 3: Calculate the force of static friction (F_static)
The force of static friction (F_static) is equal to the gravitational force acting along the ramp, which can be calculated using the formula:
F_gravity_parallel = W * sin(θ)
Where θ is the angle of the ramp (20 degrees).
Step 4: Compare F_static and F_max_static
If F_static is less than F_max_static, the block will not start sliding on its own. However, if F_static is greater than or equal to F_max_static, the block will start sliding.
Now that we have the necessary steps, let's calculate the values:
Step 1: Calculate the normal force
Since the block is resting on a ramp, the normal force is not equal to the weight (W), but only a component of it. The normal force can be found using the formula:
N = W * cos(θ)
Where θ is the angle of the ramp (20 degrees).
Step 1a: Calculate the weight (W)
Let's assume the mass of the block is 1 kilogram.
W = m * g
W = 1 kg * 9.8 m/s²
Step 1b: Calculate the normal force (N)
N = W * cos(θ)
N = W * cos(20°)
Step 2: Calculate the maximum static friction force (F_max_static)
F_max_static = μ_static * N
Step 3: Calculate the force of static friction (F_static)
F_gravity_parallel = W * sin(θ)
Step 4: Compare F_static and F_max_static
If F_static < F_max_static, then the block will not start sliding. If F_static ≥ F_max_static, then the block will start sliding.
Once you have the values from these steps, you can determine whether the block will start sliding on its own.