A block of mass 2.20 kg is accelerated across a rough surface by a light cord passing over a small pulley as shown in the figure below. The tension T in the cord is maintained at 10.0 N, and the pulley is 0.120 m above the top of the block. The coefficient of kinetic friction is 0.400.
ma=Tcos(theta) + mu(mg-tsin(theta))
Why did the block go to the therapist? Because it had too much friction with the surface and needed some counseling!
To find the acceleration of the block, we can use Newton's second law of motion. The net force acting on the block can be calculated by subtracting the force of friction from the tension in the cord.
1. Calculate the force of friction:
The force of friction can be found using the equation:
F_friction = μ * N
Where:
- μ is the coefficient of kinetic friction.
- N is the normal force, which is equal to the weight of the block.
The weight of the block can be calculated using:
Weight = mass * gravitational acceleration
Given:
- μ = 0.400 (coefficient of kinetic friction)
- mass = 2.20 kg
- gravitational acceleration = 9.8 m/s^2
Calculate the weight:
Weight = 2.20 kg * 9.8 m/s^2
2. Determine the 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, the normal force is equal in magnitude but opposite in direction to the weight.
Normal force = Weight
3. Calculate the force of friction:
F_friction = 0.400 * Normal force
4. Calculate the net force:
Net force = Tension - F_friction
Given:
- Tension (T) = 10.0 N (maintained by the cord)
5. Calculate the acceleration:
According to Newton's second law, the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
Net force = mass * acceleration
Rearranging the equation, we get:
acceleration = Net force / mass
Calculate the acceleration:
acceleration = Net force / 2.20 kg
Now, using the above steps, we can calculate the acceleration of the block.
To find the acceleration of the block across the rough surface, we can use Newton's second law of motion. The equation is given by:
ΣF = m * a
where ΣF is the net force acting on the block, m is the mass of the block, and a is the acceleration of the block.
First, we need to find the net force acting on the block. The net force is the vector sum of all the forces acting on the block. In this case, the forces acting on the block are the tension force (T) and the force of friction (F_friction).
The tension force (T) in the cord is given as 10.0 N.
The force of friction (F_friction) can be calculated using the formula:
F_friction = μ * N
where μ is the coefficient of kinetic friction and N is the normal force.
The normal force (N) acting on the block is equal to the weight of the block (m * g), where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Calculating N: N = m * g = 2.20 kg * 9.8 m/s^2 = 21.56 N
Now, we can calculate F_friction: F_friction = 0.400 * 21.56 N = 8.624 N
The net force acting on the block is given by: ΣF = T - F_friction
Substituting the given values in the equation: ΣF = 10.0 N - 8.624 N = 1.376 N
Now, we can use Newton's second law to find the acceleration of the block:
1.376 N = 2.20 kg * a
Solving for a: a = 1.376 N / 2.20 kg = 0.625 m/s^2
Therefore, the acceleration of the block across the rough surface is 0.625 m/s^2.