In this problem, we have a heavy ball suspended from the ceiling of a motor car via a light string. The speed of a transverse pulse on the string changes depending on whether the car is at rest or accelerating. We are tasked with finding the acceleration of the car based on the change in pulse speed.
Key Concepts Involved
The speed of a transverse wave on a string depends on the tension in the string, which, in turn, is affected by the force acting on the string. The tension in the string when the car is accelerating is different from when it is at rest because the acceleration of the car causes an additional force on the string.
Step 1: Understanding the change in wave speed
The speed of a transverse wave on a string is given by the equation:
v = √(T / μ)
Where:
- v is the speed of the wave on the string
- T is the tension in the string
- μ is the mass per unit length of the string
Step 2: Effect of acceleration on tension
When the car is at rest, the tension in the string is simply due to the weight of the ball hanging vertically:
T₀ = m·g
Where:
- m is the mass of the ball
- g is the acceleration due to gravity (approximately 9.8 m/s²)
However, when the car accelerates, the tension in the string increases because the ball experiences an additional horizontal force due to the car's acceleration. The total tension in the string when the car accelerates is given by:
T = m·(g + a)
Where a is the acceleration of the car.
Step 3: Relating wave speeds to tension
The speed of the wave on the string changes with the tension. Since the speed of the wave increases when the car accelerates, we can write the relationship for the speeds as:
(v₁ / v₀) = √[(T / T₀)] = √[(g + a) / g]
Where:*
v₀ is the speed of the wave when the car is at rest (50 cm/s),*
v₁ is the speed of the wave when the car accelerates (52 cm/s),*
g is the acceleration due to gravity,*
a is the acceleration of the car, which we need to find.
Step 4: Plugging in the values
We know the following values:- v₀ = 50 cm/s = 0.50 m/s,- v₁ = 52 cm/s = 0.52 m/s,- g = 9.8 m/s².Substitute these into the formula:
(0.52 / 0.50) = √[(9.8 + a) / 9.8]
Square both sides to eliminate the square root:
(0.52 / 0.50)² = (9.8 + a) / 9.8
Simplifying:
(1.04)² = (9.8 + a) / 9.8
1.0816 = (9.8 + a) / 9.8
Multiply both sides by 9.8:
1.0816 × 9.8 = 9.8 + a
10.61 = 9.8 + a
Solve for
a:
a = 10.61 - 9.8 = 0.81 m/s²
Final Answer
The acceleration of the car is approximately 0.81 m/s².
