A disc at rest of Mars 100 kg and radius 0.4 m acquire a constant angular alciliation of eight radius calculate the angular velocity in two seconds calculate angular displacement calculate linear velocity calculate tangential acceleration calculate centripetal acceleration calculate resultant acceleration and calculate centripetal Force

Question

Asked on March 18, 2026Physics

This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.

Accepted Answer · 2026-03-18T07:37:58.738Z

Step 1: Calculate the angular velocity in two seconds. The initial angular velocity is _0 = 0 rad/s. The constant angular acceleration is = 8 rad/s^2. The time is t = 2 s. Using the kinematic equation for angular velocity: = _0 + t = 0 \ rad/s + (8 \ rad/s^2)(2 \ s) = 16 \ rad/s The angular velocity in two seconds is 16 \ rad/s. Step 2: Calculate angular displacement. Using the kinematic equation for angular displacement: = _0 t + (1)/(2) t^2 = (0 \ rad/s)(2 \ s) + (1)/(2) (8 \ rad/s^2)(2 \ s)^2 = 0 + (1)/(2) (8 \ rad/s^2)(4 \ s^2) = 16 \ rad The angular displacement is 16 \ rad. Step 3: Calculate linear velocity. The radius of the disc is R = 0.4 m. The angular velocity at t=2 s is = 16 rad/s (from Step 1). Using the relation between linear and angular velocity: v = R v = (0.4 \ m)(16 \ rad/s) v = 6.4 \ m/s The linear velocity is 6.4 \ m/s. Step 4: Calculate tangential acceleration. The radius of the disc is R = 0.4 m. The constant angular acceleration is = 8 rad/s^2. Using the relation between tangential and angular acceleration: a_t = R a_t = (0.4 \ m)(8 \ rad/s^2) a_t = 3.2 \ m/s^2 The tangential acceleration is 3.2 \ m/s^2. Step 5: Calculate centripetal acceleration. The radius of the disc is R = 0.4 m. The angular velocity at t=2 s is = 16 rad/s (from Step 1). Using the formula for centripetal acceleration: a_c = R ^2 a_c = (0.4 \ m)(16 \ rad/s)^2 a_c = (0.4 \ m)(256 \ rad^2/s^2) a_c = 102.4 \ m/s^2 The centripetal acceleration is 102.4 \ m/s^2. Step 6: Calculate resultant acceleration. The tangential acceleration is a_t = 3.2 m/s^2 (from Step 4). The centripetal acceleration is a_c = 102.4 m/s^2 (from Step 5). The resultant acceleration is the vector sum of the tangential and centripetal accelerations, which are perpendicular to each other: a_res = sqrt(a_t^2 + a_c^2) a_res = sqrt((3.2 \ m/s^2))^2 + (102.4 \ m/s^2)^2 a_res = sqrt(10.24 \ m^2/s^4) + 10485.76 \ m^2/s^4 a_res = sqrt(10496 \ m^2/s^4) a_res ≈ 102.45 \ m/s^2 The resultant acceleration is 102.45 \ m/s^2. Step 7: Calculate centripetal force. The mass of the disc is m = 100 kg. The centripetal acceleration is a_c = 102.4 m/s^2 (from Step 5). Using Newton's second law for centripetal force: F_c = m a_c F_c = (100 \ kg)(102.4 \ m/s^2) F_c = 10240 \ N The centripetal force is 10240 \ N.

Accepted Answer · 2026-03-18T07:37:58.738Z

Step 1: Calculate the angular velocity in two seconds. The initial angular velocity is _0 = 0 rad/s. The constant angular acceleration is = 8 rad/s^2. The time is t = 2 s. Using the kinematic equation for angular velocity: = _0 + t = 0 \ rad/s + (8 \ rad/s^2)(2 \ s) = 16 \ rad/s The angular velocity in two seconds is 16 \ rad/s. Step 2: Calculate angular displacement. Using the kinematic equation for angular displacement: = _0 t + (1)/(2) t^2 = (0 \ rad/s)(2 \ s) + (1)/(2) (8 \ rad/s^2)(2 \ s)^2 = 0 + (1)/(2) (8 \ rad/s^2)(4 \ s^2) = 16 \ rad The angular displacement is 16 \ rad. Step 3: Calculate linear velocity. The radius of the disc is R = 0.4 m. The angular velocity at t=2 s is = 16 rad/s (from Step 1). Using the relation between linear and angular velocity: v = R v = (0.4 \ m)(16 \ rad/s) v = 6.4 \ m/s The linear velocity is 6.4 \ m/s. Step 4: Calculate tangential acceleration. The radius of the disc is R = 0.4 m. The constant angular acceleration is = 8 rad/s^2. Using the relation between tangential and angular acceleration: a_t = R a_t = (0.4 \ m)(8 \ rad/s^2) a_t = 3.2 \ m/s^2 The tangential acceleration is 3.2 \ m/s^2. Step 5: Calculate centripetal acceleration. The radius of the disc is R = 0.4 m. The angular velocity at t=2 s is = 16 rad/s (from Step 1). Using the formula for centripetal acceleration: a_c = R ^2 a_c = (0.4 \ m)(16 \ rad/s)^2 a_c = (0.4 \ m)(256 \ rad^2/s^2) a_c = 102.4 \ m/s^2 The centripetal acceleration is 102.4 \ m/s^2. Step 6: Calculate resultant acceleration. The tangential acceleration is a_t = 3.2 m/s^2 (from Step 4). The centripetal acceleration is a_c = 102.4 m/s^2 (from Step 5). The resultant acceleration is the vector sum of the tangential and centripetal accelerations, which are perpendicular to each other: a_res = sqrt(a_t^2 + a_c^2) a_res = sqrt((3.2 \ m/s^2))^2 + (102.4 \ m/s^2)^2 a_res = sqrt(10.24 \ m^2/s^4) + 10485.76 \ m^2/s^4 a_res = sqrt(10496 \ m^2/s^4) a_res ≈ 102.45 \ m/s^2 The resultant acceleration is 102.45 \ m/s^2. Step 7: Calculate centripetal force. The mass of the disc is m = 100 kg. The centripetal acceleration is a_c = 102.4 m/s^2 (from Step 5). Using Newton's second law for centripetal force: F_c = m a_c F_c = (100 \ kg)(102.4 \ m/s^2) F_c = 10240 \ N The centripetal force is 10240 \ N.

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