a bicycle has a front wheel with a diameter of 24cm and a back wheel with a diameter of 60cm. through what angle does the front wheel turn if the back wheel turns through 12 radians
distance covered by both wheels must be the same ...
with radians, arclength is simply rØ
so 24Ø = 60(12)
Ø = 30 radians
Well, if the back wheel is turning through 12 radians, then the front wheel is probably saying, "Hold my spokes, I got this!" Now, let's get down to business. The circumference of a circle is given by the formula C = 2πr, where r is the radius.
For the back wheel with a diameter of 60cm, we can find the radius by dividing the diameter by 2: r = 60cm / 2 = 30cm. So, the circumference of the back wheel is C1 = 2π × 30cm.
Now, let's move on to the front wheel with a diameter of 24cm. Using the same logic, we find the radius to be r = 24cm / 2 = 12cm. Thus, the circumference of the front wheel is C2 = 2π × 12cm.
Since the back wheel is turning through 12 radians, we can find the angle the front wheel turns through by using the formula:
Angle = (Distance traveled by front wheel) / (Radius of front wheel)
The distance traveled by the front wheel is equal to the circumference of the back wheel, as both wheels travel the same distance.
Therefore, the angle the front wheel turns through is:
Angle = C1 / r2 = (2π × 30cm) / 12cm = 5π radians.
So, the front wheel turns through an angle of 5π radians. And remember, the front wheel always follows the back wheel's lead, even when it comes to angles!
To determine the angle through which the front wheel turns, we can use a ratio based on the circumference of the wheels.
Step 1: Calculate the circumference of the front and back wheels.
The circumference of a circle is given by the formula C = πd, where C is the circumference and d is the diameter.
For the front wheel:
Circumference of front wheel = π * diameter of front wheel
Circumference of front wheel = π * 24 cm
For the back wheel:
Circumference of back wheel = π * diameter of back wheel
Circumference of back wheel = π * 60 cm
Step 2: Calculate the number of revolutions the back wheel makes.
Since the back wheel turns through 12 radians, we need to calculate the number of revolutions it makes. One revolution is equal to 2π radians.
Number of revolutions = 12 radians / (2π radians/revolution)
Step 3: Calculate the length of the distance traveled by the back wheel.
The distance traveled by the back wheel is equal to the circumference of the back wheel multiplied by the number of revolutions.
Distance traveled by back wheel = Circumference of back wheel * Number of revolutions
Step 4: Calculate the distance traveled by the front wheel.
Since the front wheel is attached to the same bicycle, it will travel an equal distance to the back wheel.
Distance traveled by front wheel = Distance traveled by back wheel
Step 5: Calculate the angle through which the front wheel turns.
The angle through which the front wheel turns can be calculated by dividing the distance traveled by the front wheel by its circumference, and multiplying by 360 degrees (or 2π radians).
Angle = (Distance traveled by front wheel / Circumference of front wheel) * 360 degrees
Alternatively, we can calculate the angle directly by dividing the radians traveled by the back wheel by the radians traveled per revolution of the front wheel, and multiplying by 360 degrees.
Angle = (12 radians / (2π radians/revolution)) * 360 degrees
Using the given values, we can substitute them into the formula to find the angle through which the front wheel turns.
To find the angle by which the front wheel turns, we'll use the concept of angular displacement.
The angular displacement (θ) refers to the angle through which an object rotates. In this case, we'll find the angular displacement of the front wheel based on the given angular displacement of the back wheel.
We know that the angle turned by a wheel is directly proportional to the ratio of the distance traveled by a point on the circumference of the wheel to the circumference of the wheel.
The distance traveled by the point on the circumference of the back wheel can be calculated using its circumference:
Distance traveled by the back wheel = Circumference of the back wheel × angular displacement of the back wheel
Distance traveled by the back wheel = 2π × (radius of the back wheel)
Distance traveled by the back wheel = 2π × (diameter of the back wheel / 2)
Distance traveled by the back wheel = π × (diameter of the back wheel)
Similarly, the distance traveled by the point on the circumference of the front wheel would be:
Distance traveled by the front wheel = Circumference of the front wheel × angular displacement of the front wheel
Distance traveled by the front wheel = 2π × (radius of the front wheel)
Distance traveled by the front wheel = 2π × (diameter of the front wheel / 2)
Distance traveled by the front wheel = π × (diameter of the front wheel)
Since the distances traveled by both wheels must be equal (assuming both wheels roll without slipping), we can set up the equation:
Distance traveled by the back wheel = Distance traveled by the front wheel
π × (diameter of the back wheel) = π × (diameter of the front wheel)
Using the given diameters:
π × 60 = π × (24 × θ) [angular displacement of the front wheel = θ]
Simplifying the equation:
60 = 24 × θ
θ = 60 / 24
θ = 2.5 radians
Therefore, the front wheel turns through 2.5 radians when the back wheel turns through 12 radians.