A disk with a mass of 16 kg and a radius of 41 cm is mounted on a frictionless horizontal axle. A string is wound many times around the disk and then attached to a 70-kg block, as shown in the figure. Find the acceleration of the block, assuming that the string does not slip.
To find the acceleration of the block, we will use Newton's second law of motion which states that the net force acting on an object is equal to the product of its mass and acceleration.
Let's break down the forces acting on the system:
1. Tension in the string: The tension in the string will provide the force to accelerate the block.
2. Force of friction: Since the axle is frictionless, there is no friction force acting on the disk.
For the block:
Since the tension in the string is the only force acting on the block, we can write the equation as follows:
Tension = mass of the block × acceleration of the block
For the disk:
The tension in the string is also responsible for the rotational motion of the disk. The torque (τ) produced by the tension provides the angular acceleration (α) of the disk. Since there is no friction, the torque is equal to the tension multiplied by the radius of the disk.
τ = Tension × radius of the disk
We can also relate torque (τ) and angular acceleration (α) using the formula:
τ = moment of inertia (I) × angular acceleration (α)
Now, let's determine the moment of inertia (I) of the disk:
The moment of inertia of a solid disk rotating about its axis is given by the formula:
I = (1/2) × mass of the disk × radius of the disk²
After determining the angular acceleration (α) of the disk, we can relate it to the linear acceleration (a) of the block by the formula:
a = α × radius of the disk
Now let's solve the problem step-by-step:
Step 1: Calculate the tension in the string:
The tension in the string (Tension) can be determined by equating it to mass of the block (70 kg) × acceleration of the block (a). This gives us:
Tension = 70 kg × a
Step 2: Calculate the moment of inertia (I) of the disk:
The moment of inertia (I) of the disk can be calculated using the formula:
I = (1/2) × mass of the disk × radius of the disk²
I = (1/2) × 16 kg × (0.41 m)²
Step 3: Relate torque (τ) and angular acceleration (α) of the disk:
Since torque (τ) is equal to the tension (Tension) × radius of the disk, we can write:
τ = Tension × radius of the disk
Substituting the value of Tension from Step 1 and the radius of the disk, we get:
τ = (70 kg × a) × 0.41 m
Step 4: Determine the angular acceleration (α) of the disk:
Using the formula τ = I × α and substituting the value of τ from Step 3 and the value of I from Step 2, we have:
(70 kg × a) × 0.41 m = (1/2) × 16 kg × (0.41 m)² × α
Step 5: Relate the angular acceleration (α) of the disk to the linear acceleration (a) of the block:
Using the formula a = α × radius of the disk and substituting the value of the radius of the disk, we find:
a = α × 0.41 m
Step 6: Finally, solve for the acceleration of the block (a):
By equating the expressions for α from Step 4 and Step 5, we can solve for a:
(70 kg × a) × 0.41 m = (1/2) × 16 kg × (0.41 m)² × α
70a = (1/2) × 16 × 0.41² × α
Simplifying the equation, we find:
a = (1/2) × (0.41 m)² × α
Since α = a / (0.41 m), we can substitute this value into the equation to get:
a = (1/2) × (0.41 m)² × (a / (0.41 m))
a = (1/2) × 0.41 m × a
a = 0.205 m/s²
Therefore, the acceleration of the block is 0.205 m/s².