To balance a 40.8-kg automobile tire and wheel, a mechanic must place a 41.7-g lead weight 25.0 cm from the center of the wheel. When the wheel is balanced, its center of mass is exactly at the center of the wheel. How far from the center of the wheel was its center of mass before the lead weight was added?
Call the unknown distance x.
x*40.8 kg = 0.00417 kg*25 cm
Solve for x in units of centimeters
**41.7g = 0.0417kg
not, 0.00417 =)
To find the distance from the center of the wheel to its center of mass before the lead weight was added, we can use the principle of moments or torque.
Torque is a measure of the turning force on an object and is calculated by multiplying the force by the distance from the pivot point. In this case, the pivot point is the center of the wheel.
Before the lead weight was added, the total torque was zero, as the center of mass was at the center of the wheel. When the lead weight was added, it created an unbalanced torque that needs to be balanced by adjusting the position of the center of mass.
The torque due to the lead weight is given by:
τ = force × distance
Here, the force is the weight of the lead (41.7 g) and the distance is the distance from the center of the wheel (unknown) to the lead weight (25.0 cm).
Since torque is a rotational force, we need to convert the weight of the lead from grams to newtons. The conversion factor is 1 g = 0.0098 N.
Therefore, the torque due to the lead weight is:
τ = (41.7 g × 0.0098 N/g) × 25.0 cm
Simplifying this expression gives:
τ = 41.09 cm·N
To balance the torque and find the distance from the center of the wheel to its center of mass before the lead weight was added, we assume the tire and wheel are balanced about their own center of mass. Thus, the torque due to the automobile tire and wheel is equal and opposite to the torque due to the lead weight:
τ_tire = -τ_lead
The negative sign indicates that the torques are in opposite directions.
Substituting the known values, we have:
(d_tire)(m_tire) = -(25.0 cm)(41.7 g × 0.0098 N/g)
Here, (d_tire) is the distance from the center of the wheel to its center of mass before the lead weight was added and (m_tire) is the mass of the tire (40.8 kg).
Rearranging the equation to solve for (d_tire), we get:
(d_tire) = -[(25.0 cm)(41.7 g × 0.0098 N/g)] / (m_tire)
Substituting the known values and converting the mass of the tire to grams to match the units, we get:
(d_tire) = -[(25.0 cm)(41.7 g × 0.0098 N/g)] / (40.8 kg × 1000 g/kg)
Simplifying the expression gives:
(d_tire) = -0.030 m
Therefore, the center of mass of the wheel was 0.030 meters (30 cm) from the center of the wheel before the lead weight was added.