A car has wheels with tyres on, which are 75cm in diameter. The car is travelling at a constant speed of 60km/hr. If no slipping is observed between the wheels(tyres) and the road, calculate the angular speed and the acceleration of a stone wedged in the thread of the tyre
V = 60,000/3600s = 16.7 m/s.
C = pi*D = 3.14 * 0.75m = 2.36 m.
Va = 16.7m/s * 6.28Rad/2.36m=44.3 Rad/s.
To calculate the angular speed of the stone, we first need to find the circumference of the tire:
Circumference = π * diameter
Circumference = π * 75 cm
Circumference ≈ 235.62 cm
Since the car is moving at a constant speed of 60 km/hr, we need to convert it to cm/s:
Speed = 60 km/hr = (60 * 1000) m/3600 s = 1666.67 cm/s
Now, we can calculate the angular speed:
Angular speed = Speed / Circumference
Angular speed = 1666.67 cm/s / 235.62 cm
Angular speed ≈ 7.07 rad/s
To calculate the acceleration of the stone, we need to consider that the stone is moving in a circle around the center of the wheel. The acceleration can be found using the centripetal acceleration formula:
Centripetal acceleration = (Angular speed)^2 * Radius
Since we know the diameter of the wheel (75 cm), we can find the radius:
Radius = Diameter / 2
Radius = 75 cm / 2
Radius = 37.5 cm
Now we can calculate the acceleration:
Centripetal acceleration = (7.07 rad/s)^2 * 37.5 cm
Centripetal acceleration ≈ 1875 cm/s^2
Therefore, the angular speed of the stone is approximately 7.07 rad/s, and the acceleration of the stone is approximately 1875 cm/s^2.
To calculate the angular speed and acceleration of the stone wedged in the tread of the tire, we will need to break down the problem step by step.
1. First, let's convert the car's speed from kilometers per hour to meters per second since angular speed is usually measured in radians per second. We can multiply 60 km/hr by 1000/3600 to get the speed in meters per second. This gives us:
Speed = (60 km/hr) * (1000 m/1 km) * (1 hr/3600 s) = 16.67 m/s
2. To find the angular speed of the tire, we need to understand that one complete revolution of the tire corresponds to an angle of 2π radians. The distance traveled by the tire in one revolution can be calculated using the circumference formula:
Circumference = π * diameter
For a tire with a diameter of 75 cm, the circumference would be:
Circumference = π * 75 cm = 235.62 cm
We need to convert this circumference to meters:
Circumference = 235.62 cm * (1 m/100 cm) = 2.3562 m
Therefore, in one revolution, the tire travels a distance of 2.3562 m.
3. Now, we can calculate the angular speed using the formula:
Angular Speed = Speed / Distance
Angular Speed = 16.67 m/s / 2.3562 m ≈ 7.08 radians/s
Therefore, the angular speed of the tire is approximately 7.08 radians/s.
4. To find the acceleration of the stone, we need to consider that the stone is rotating with the same angular speed as the tire. The acceleration of an object moving in a circle depends on a few factors, including the radius of the circle and the angular speed. Since the stone is wedged in the tread of the tire, its radius can be considered as the radius of the tire itself.
Acceleration = (Angular Speed)^2 * Radius
The radius of the tire can be calculated by dividing its diameter by 2:
Radius = 75 cm / 2 = 37.5 cm
We convert this radius to meters:
Radius = 37.5 cm * (1 m/100 cm) = 0.375 m
Now we can calculate the acceleration:
Acceleration = (7.08 radians/s)^2 * 0.375 m ≈ 18.76 m/s^2
Therefore, the acceleration of the stone wedged in the tire tread is approximately 18.76 m/s^2.